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Mathematics LibreTexts

9.8: Rational Exponents

  • Page ID
    18994
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    Skills to Develop

    By the end of this section, you will be able to:
    • Simplify expressions with \(a^{\frac{1}{n}}\)
    • Simplify expressions with \(a^{\frac{m}{n}}\)
    • Use the Laws of Exponents to simply expressions with rational exponents

    Before you get started, take this readiness quiz.

    1. Add: \(\frac{7}{15}+\frac{5}{12}\).
      If you missed this problem, review [link].
    2. Simplify: \((4x^{2}y^{5})^3\).
      If you missed this problem, review [link].
    3. Simplify: \(5^{−3}\).
      If you missed this problem, review [link].

    Simplify Expressions with \(a^{\frac{1}{n}}\)

    Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

    The Power Property for Exponents says that \((a^m)^n=a^{m·n}\) when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

    Suppose we want to find a number p such that \((8^p)^3=8\). We will use the Power Property of Exponents to find the value of p.

    \[\begin{array}{cc} {}&{(8^p)^3=8}\\ {\text{Multiply the exponents on the left.}}&{8^{3p}=8}\\ {\text{Write the exponent 1 on the right.}}&{8^{3p}=8^1}\\ {\text{The exponents must be equal.}}&{3p=1}\\ {\text{Solve for p.}}&{p=\frac{1}{3}}\\ \nonumber \end{array}\]

    But we know also \((\sqrt[3]{8})^3=8\). Then it must be that \(8^{\frac{1}{3}}=\sqrt[3]{8}\)

    This same logic can be used for any positive integer exponent n to show that \(a^{\frac{1}{n}}=\sqrt[n]{a}\).

    Definition: RATIONAL EXPONENT \(a^{\frac{1}{n}}\)

    If \(\sqrt[n]{a}\) is a real number and \(n \ge 2\), \(a^{\frac{1}{n}}=\sqrt[n]{a}\).

    There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

    Example \(\PageIndex{1}\)

    Write as a radical expression:

    1. \(x^{\frac{1}{2}}\)
    2. \(y^{\frac{1}{3}}\)
    3. \(z^{\frac{1}{4}}\).
    Answer

    We want to write each expression in the form \(\sqrt[n]{a}\).

    1. \(x^{\frac{1}{2}}\)
    The denominator of the exponent is 2, so the index of the radical is 2. We do not show the index when it is 2. \(\sqrt{x}\)
    2. \(y^{\frac{1}{3}}\)
    The denominator of the exponent is 3, so the index is 3. \(\sqrt[3]{y}\)
    3. \(z^\frac{1}{4}}\)
    The denominator of the exponent is 4, sothe index is 4. \(\sqrt[4]{z}\)

    Example \(\PageIndex{2}\)

    Write as a radical expression:

    1. \(t^{\frac{1}{2}}\)
    2. \(m^{\frac{1}{3}}\)
    3. \(r^{\frac{1}{4}}\).
    Answer
    1. \(\sqrt{t}\)
    2. \(\sqrt[3]{m}\)
    3. \(\sqrt[4]{r}\)

    Example \(\PageIndex{3}\)

    Write as a radical expression:

    1. \(b^{\frac{1}{2}}\)
    2. \(z^{\frac{1}{3}}\)
    3. \(p^{\frac{1}{4}}\).
    Answer
    1. \(\sqrt{b}\)
    2. \(\sqrt[3]{z}\)
    3. \(\sqrt[4]{p}\)

    Example \(\PageIndex{4}\)

    Write with a rational exponent:

    1. \(\sqrt{x}\)
    2. \(\sqrt[3]{y}\)
    3. \(\sqrt[4]{z}\).
    Answer

    We want to write each radical in the form \(a^{\frac{1}{n}}\).

    1. \(\sqrt{x}\)
    No index is shown, so it is 2. The denominator of the exponent will be 2. \(x^{\frac{1}{2}}\)
    2. \(\sqrt[3]{y}\)
    The index is 3, so the denominator of the exponent is 3. \(y^{\frac{1}{3}}\)
    3. \(\sqrt[4]{z}\)
    The index is 4, so the denominator of the exponent is 4. \(z^{\frac{1}{4}}\)

    Example \(\PageIndex{5}\)

    Write with a rational exponent:

    1. \(\sqrt{s}\)
    2. \(\sqrt[3]{x}\)
    3. \(\sqrt[4]{b}\).
    Answer
    1. \(s^{\frac{1}{2}}\)
    2. \(x^{\frac{1}{3}}\)
    3. \(b^{\frac{1}{4}}\

    Example \(\PageIndex{6}\)

    Write with a rational exponent:

    1. \(\sqrt{v}\)
    2. \(\sqrt[3]{p}\)
    3. \(\sqrt[4]{p}\).
    Answer
    1. \(v^{\frac{1}{2}}\)
    2. \(p^{\frac{1}{3}}\)
    3. \(p^{\frac{1}{4}}\)

    Example \(\PageIndex{7}\)

    Write with a rational exponent:

    1. \(\sqrt{5y}\)
    2. \(\sqrt[3]{4x}\)
    3. \(3\sqrt[4]{5z}\).
    Answer
    1. \(\sqrt{5y}\)
    No index is shown, so it is 2. The denominator of the exponent will be 2. \((5y)^{\frac{1}{2}}\)
    2. \(\sqrt[3]{4x}\)
    The index is 3, so the denominator of the exponent is 3. \((4x)^{\frac{1}{3}}\)
    3. \(3\sqrt[4]{5z}\)
    The index is 4, so the denominator of the exponent is 4. \(3(5z)^{\frac{1}{4}}\)

    Example \(\PageIndex{8}\)

    Write with a rational exponent:

    1. \(\sqrt{10m}\)
    2. \(\sqrt[5]{3n}\)
    3. \(3\sqrt[4]{6y}\).
    Answer
    1. \((10^m)^{\frac{1}{2}}\)
    2. \((3n)^{\frac{1}{5}}\)
    3. \((486y)^{\frac{1}{4}}\)

    Example \(\PageIndex{9}\)

    Write with a rational exponent:

    1. \(\sqrt[7]{3k}\)
    2. \(\sqrt[4]{5j}\)
    3. \(\sqrt[3]{82a}\).
    Answer
    1. \((3k)^{\frac{1}{7}}\)
    2. \((5j)^{\frac{1}{4}}\)
    3. \((1024a)^{\frac{1}{3}}\)

    In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

    Example \(\PageIndex{10}\)

    Simplify:

    1. \(25^{\frac{1}{2}}\)
    2. \(64^{\frac{1}{3}}\)
    3. \(256^{\frac{1}{4}}\).
    Answer
    1. \(25^{\frac{1}{2}}\)
    Rewrite as a square root. \(\sqrt{25}\)
    Simplify. 5
    2. \(64^{\frac{1}{3}}\)
    Rewrite as a cube root. \(\sqrt[3]{64}\)
    Recognize 64 is a perfect cube. \(\sqrt[3]{4^3}\)
    Simplify. 4
    3. \(256^{\frac{1}{4}}\)
    Rewrite as a fourth root. \(\sqrt[4]{256}\)
    Recognize 256 is a perfect fourth power. \(\sqrt[4]{4^4}\)
    Simplify. 4

    Example \(\PageIndex{11}\)

    Simplify:

    1. \(36^{\frac{1}{2}}\)
    2. \(8^{\frac{1}{3}}\)
    3. \(16^{\frac{1}{4}}\).
    Answer
    1. 6
    2. 2
    3. 2

    Example \(\PageIndex{12}\)

    Simplify:

    1. \(100^{\frac{1}{2}}\)
    2. \(27^{\frac{1}{3}}\)
    3. \(81^{\frac{1}{4}}\).
    Answer
    1. 10
    2. 3
    3. 3

    Be careful of the placement of the negative signs in the next example. We will need to use the property \(a^{−n}=\frac{1}{a^n}\) in one case.

    Example \(\PageIndex{13}\)

    Simplify:

    1. \((−64)^{\frac{1}{3}}\)
    2. \(−64^{\frac{1}{3}}\)
    3. \((64)^{−\frac{1}{3}}\).
    Answer
    1. \((−64)^{\frac{1}{3}}\)
    Rewrite as a cube root. \(\sqrt[3]{−64}\)
    Rewrite−64 as a perfect cube. \(\sqrt[3]{(−4)^3}\)
    Simplify. −4
    2. \(−64^{\frac{1}{3}}\)
    The exponent applies only to the 64. \(−(64^{\frac{1}{3}})\)
    Rewrite as a cube root. \(−\sqrt[3]{64}\)
    Rewrite 64 as \(4^3\). \(−\sqrt[3]{4^3}\)
    Simplify. −4
    3. \((64)^{−\frac{1}{3}}\)

    Rewrite as a fraction with a positive exponent, using the property, \(a^{−n}=\frac{1}{a^n}\).

    Write as a cube root.

    \(\frac{1}{\sqrt[3]{64}}\)
    Rewrite 64 as \(4^3\). \(\frac{1}{\sqrt[3]{4^3}}\)
    Simplify. \(\frac{1}{4}\)

    Example \(\PageIndex{14}\)

    Simplify:

    1. \((−125)^{\frac{1}{3}}\)
    2. \(−125^{\frac{1}{3}}\)
    3. \((125)^{−\frac{1}{3}}\).
    Answer
    1. −5
    2. −5
    3. \(\frac{1}{5}\)

    Example \(\PageIndex{15}\)

    Simplify:

    1. \((−32)^{\frac{1}{5}}\)
    2. \(−32^{\frac{1}{5}}\)
    3. \((32)^{−\frac{1}{5}}\).
    Answer
    1. −2
    2. −2
    3. \(\frac{1}{2}\)

    Example \(\PageIndex{16}\)

    Simplify:

    1. \((−16)^{\frac{1}{4}}\)
    2. \(−16^{\frac{1}{4}}\)
    3. \((16)^{−\frac{1}{4}}\).
    Answer
    1. \((−16)^{\frac{1}{4}}\)
    Rewrite as a fourth root. \(\sqrt[4]{−16}\)
    There is no real number whose fourth power is −16.
    2. \(−16^{\frac{1}{4}}\)
    The exponent applies only to the 16. \(−(16^{\frac{1}{4}})\)
    Rewrite as a fourth root. \(−\sqrt[4]{16}\)
    Rewrite 16 as \(2^4\) \(−\sqrt[4]{2^4}\)
    Simplify. −2
    3. \((16)^{−\frac{1}{4}}\)

    Rewrite as a fraction with a positive exponent, using the property, \(a^{−n}=\frac{1}{a^n}\).

    \(\frac{1}{(16)^{\frac{1}{4}}}\)
    Rewrite as a fourth root. \(\frac{1}{\sqrt[4]{16}}\)
    Rewrite 16 as \(2^4\). \(\frac{1}{\sqrt[4]{2^4}}\)
    Simplify. \(\frac{1}{2}\)

    Example \(\PageIndex{17}\)

    Simplify:

    1. \((−64)^{\frac{1}{2}}\)
    2. \(−64^{\frac{1}{2}}\)
    3. \((64)^{−\frac{1}{2}}\).
    Answer
    1. −8
    2. −8
    3. \(\frac{1}{8}\)

    Example \(\PageIndex{18}\)

    Simplify:

    1. \((−256)^{\frac{1}{4}}\)
    2. \(−256^{\frac{1}{4}}\)
    3. \((256)^{−\frac{1}{4}}\).
    Answer
    1. −4
    2. −4
    3. \(\frac{1}{4}\)

    Simplify Expressions with \(a^{\frac{m}{n}}\)

    Let’s work with the Power Property for Exponents some more.

    Suppose we raise \(a^{\frac{1}{n}}\) to the power m.

    \[\begin{array}{ll} {}&{(a^{\frac{1}{n}})^m}\\ {\text{Multiply the exponents.}}&{a^{\frac{1}{n}·m}}\\ {\text{Simplify.}}&{a^{\frac{m}{n}}}\\ {\text{So} a^{\frac{m}{n}}=(\sqrt[n]{a})^m \text{also.}}&{}\\ \nonumber \end{array}\]

    Now suppose we take \(a^m\) to the \(\frac{1}{n}\) power.

    \[\begin{array}{ll} {}&{(a^m)^{\frac{1}{n}}}\\ {\text{Multiply the exponents.}}&{a^{m·\frac{1}{n}}}\\ {\text{Simplify.}}&{a^{\frac{m}{n}}}\\ {\text{So} a^{\frac{m}{n}}=\sqrt[n]{a^m} \text{also.}}&{}\\ \nonumber \end{array}\]

    Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller.

    Definition: RATIONAL EXPONENT \(a^{\frac{m}{n}}\)

    For any positive integers m and n,

    \(a^{\frac{m}{n}}=(\sqrt[n]{a})^m\)

    \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)

    Example \(\PageIndex{19}\)

    Write with a rational exponent:

    1. \(\sqrt{y^3}\)
    2. \(\sqrt[3]{x^2}\)
    3. \(\sqrt[4]{z^3}\)
    Answer

    We want to use \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\) to write each radical in the form \(a^{\frac{m}{n}}\).


    1. This figure says, “The numerator of the exponent is the exponent of y, 3.” It then shows the square root of y cubed. The figure then says, “The denominator of the exponent is the index of the radical, 2.” It then shows y to the 3/2 power.

    2. This figure says, “The numerator of the exponent is the exponent of x, 2.” It then shows the cubed root of x squared. The figure then reads, “The denominator of the exponent is the index of the radical, 3.” It then shows y to the 2/3 power.

    3. This figure reads, “The numerator of the exponent is the exponent of z, 3.” It then shows the fourth root of z cubed. The figure then reads, “The denominator of the exponent is the index of the radical, 4.” It then shows z to the 3/4 power.

    Example \(\PageIndex{20}\)

    Write with a rational exponent:

    1. \(\sqrt{x^5}\)
    2. \(\sqrt[4]{z^3}\)
    3. \(\sqrt[5]{y^2}\).
    Answer
    1. \(x^{\frac{5}{2}}\)
    2. \(z^{\frac{3}{4}}\)
    3. \(y^{\frac{2}{5}}\)

    Example \(\PageIndex{21}\)

    Write with a rational exponent:

    1. \(\sqrt[5]{a^2}\)
    2. \(\sqrt[3]{b^7}\)
    3. \(\sqrt[4]{m^5}\).
    Answer
    1. \(a^{\frac{2}{5}}\)
    2. \(b^{\frac{7}{3}}\)
    3. \(m^{\frac{5}{4}}\)

    Example \(\PageIndex{22}\)

    Simplify:

    1. \(9^{\frac{3}{2}}\)
    2. \(125^{\frac{2}{3}}\)
    3. \(81^{\frac{3}{4}}\).
    Answer

    We will rewrite each expression as a radical first using the property, \(a^{\frac{m}{n}}=(\sqrt[n]{a})^m\). This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

    1. \(9^{\frac{3}{2}}\)
    The power of the radical is the numerator of the exponent, 3. Since the denominator of the exponent is 2, this is a square root. \((\sqrt{9})^3\)
    Simplify. \(3^3\)
    27
    2. \(125^{\frac{2}{3}}\)
    The power of the radical is the numerator of the exponent, 2. Since the denominator of the exponent is 3, this is a square root. \((\sqrt[3]{125})^2\)
    Simplify. \(5^2\)
    25
    3. \(81^{\frac{3}{4}}\)
    The power of the radical is the numerator of the exponent, 2. Since the denominator of the exponent is 3, this is a square root. \((\sqrt[4]{81})^3\)
    Simplify. \(3^3\)
    27

    Example \(\PageIndex{23}\)

    Simplify:

    1. \(4^{\frac{3}{2}}\)
    2. \(27^{\frac{2}{3}}\)
    3. \(625^{\frac{3}{4}}\).
    Answer
    1. 8
    2. 9
    3. 125

    Example \(\PageIndex{24}\)

    Simplify:

    1. \(8^{\frac{5}{3}}\)
    2. \(81^{\frac{3}{2}}\)
    3. \(16^{\frac{3}{4}}\).
    Answer
    1. 32
    2. 729
    3. 8

    Remember that \(b^{−p}=\frac{1}{b^p}\). The negative sign in the exponent does not change the sign of the expression.

    Example \(\PageIndex{25}\)

    Simplify:

    1. \(16^{−\frac{3}{2}}\)
    2. \(32^{−\frac{2}{5}}\)
    3. \(4^{−\frac{5}{2}}\)
    Answer

    We will rewrite each expression first using \(b^{−p}=\frac{1}{b^p}\) and then change to radical form.

    1. \(16^{−\frac{3}{2}}\)
    Rewrite using \(b^{−p}=\frac{1}{b^p}\). \(\frac{1}{16^{\frac{3}{2}}}\)
    Change to radical form. The power of the radical is the numerator of the exponent, 3. The index is the denominator of the exponent, 2. \(\frac{1}{(\sqrt{16})^3}\)
    Simplify. \(\frac{1}{4^3}\)
    \(\frac{1}{64}\)
    2. \(32^{−\frac{2}{5}}\)
    Rewrite using \(b^{−p}=\frac{1}{b^p}\). \(\frac{1}{32^{\frac{2}{5}}}\)
    Change to radical form. \(\frac{1}{(\sqrt[5]{32})^2}\)
    Rewrite the radicand as a power. \(\frac{1}{(\sqrt[5]{2^5})^2}\)
    Simplify. \(\frac{1}{2^2}\)
    \(\frac{1}{4}\)
    3. \(4^{−\frac{5}{2}}\)
    Rewrite using \(b^{−p}=\frac{1}{b^p}\). \(\frac{1}{4^{\frac{5}{2}}}\)
    Change to radical form. \(\frac{1}{(\sqrt{4})^5}\)
    Simplify. \(\frac{1}{2^5}\)
    \(\frac{1}{32}\)

    Example \(\PageIndex{26}\)

    Simplify:

    1. \(8^{−\frac{5}{3}}\)8
    2. \(81^{−\frac{3}{2}}\)
    3. \(16^{−\frac{3}{4}}\).
    Answer
    1. \(\frac{1}{32}\)
    2. \(\frac{1}{729}\)
    3. \(\frac{1}{8}\)

    Example \(\PageIndex{27}\)

    Simplify:

    1. \(4^{−\frac{3}{2}}\)
    2. \(27^{−\frac{2}{3}}\)
    3. \(625^{−\frac{3}{4}}\).
    Answer
    1. \(\frac{1}{8}\)
    2. \(\frac{1}{9}\)
    3. \(\frac{1}{125}\)

    Example \(\PageIndex{28}\)

    Simplify:

    1. \(−25^{\frac{3}{2}}\)
    2. \(−25^{−\frac{3}{2}}\)
    3. \((−25)^{\frac{3}{2}}\).
    Answer
    1. \(−25^{\frac{3}{2}}\)
    Rewrite in radical form. \(−(\sqrt{25})^3\)
    Simplify the radical \(−5^3\)
    Simplify. −125
    2. \(−25^{−\frac{3}{2}}\)
    Rewrite using \(b^{−p}=\frac{1}{b^p}\). \(−(\frac{1}{25^{\frac{3}{2}}})\)
    Rewrite in radical form. \(−(\frac{1}{(\sqrt{25})^3})\)
    Simplify the radical. \(−(\frac{1}{5^3})\)
    Simplify. \(−\frac{1}{125}\)
    3. \((−25)^{\frac{3}{2}}\).
    Rewrite in radical form. \((\sqrt{−25})^3\)
    There is no real number whose square root is−25. Not a real number.

    Example \(\PageIndex{29}\)

    Simplify:

    1. \(−16^{\frac{3}{2}}\)
    2. \(−16^{−\frac{3}{2}}\)
    3. \((−16)^{−\frac{3}{2}}\).
    Answer
    1. −64
    2. \(−\frac{1}{64}\)
    3. not a real number

    Example \(\PageIndex{30}\)

    Simplify:

    1. \(−81^{\frac{3}{2}}\)
    2. \(−81^{−\frac{3}{2}}\)
    3. \((−81)^{−\frac{3}{2}}\).
    Answer
    1. −729
    2. \(−\frac{1}{729}\)
    3. not a real number

    Use the Laws of Exponents to Simplify Expressions with Rational Exponents

    The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.

    SUMMARY OF EXPONENT PROPERTIES

    If a,b are real numbers and m,n are rational numbers, then

    \[\begin{array}{ll} {\textbf{Product Property}}&{a^m·a^n=a^{m+n}}\\ {\textbf{Power Property}}&{(a^m)^n=a^{m·n}}\\ {\textbf{Product to a Power}}&{(ab)^m=a^{m}b^{m}}\\ {\textbf{Quotient Property}}&{\frac{a^m}{a^n}=a^{m−n} , a \ne 0, m>n}\\ {}&{\frac{a^m}{a^n}=\frac{1}{a^{n−m}}, a \ne 0, n>m}\\ {\textbf{Zero Exponent Definition}}&{a^0=1, a \ne 0}\\ {\textbf{Quotient to a Power Property}}&{(\frac{a}{b})^m=\frac{a^m}{b^m}, b \ne 0}\\ \nonumber \end{array}\]

    When we multiply the same base, we add the exponents.

    Example \(\PageIndex{31}\)

    Simplify:

    1. \(2^{\frac{1}{2}}·2^{\frac{5}{2}}\)
    2. \(x^{\frac{2}{3}}·x^{\frac{4}{3}}\)
    3. \(z^{\frac{3}{4}}·z^{\frac{5}{4}}\).
    Answer
    1. \(2^{\frac{1}{2}}·2^{\frac{5}{2}}\)
    The bases are the same, so we add the exponents. \(2^{\frac{1}{2}+\frac{5}{2}}\)
    Add the fractions. \(2^{\frac{6}{2}}\)
    Simplify the exponent. \(2^3\)
    Simplify. 8
    2. \(x^{\frac{2}{3}}·x^{\frac{4}{3}}\)
    The bases are the same, so we add the exponents. \(x^{\frac{2}{3}+\frac{4}{3}}\)
    Add the fractions. \(x^{\frac{6}{3}}\)
    Simplify. \(x^2\)
    3. \(z^{\frac{3}{4}}·z^{\frac{5}{4}}\)
    The bases are the same, so we add the exponents. \(z^{\frac{3}{4}+\frac{5}{4}}\)
    Add the fractions. \(z^{\frac{8}{4}}\)
    Simplify. \(z^2\)

    Example \(\PageIndex{32}\)

    Simplify:

    1. \(3^{\frac{2}{3}}·3^{\frac{4}{3}}\)
    2. \(y^{\frac{1}{3}}·y^{\frac{8}{3}}\)
    3. \(m^{\frac{1}{4}}·m^{\frac{3}{4}}\).
    Answer
    1. 9
    2. \(y^3\)
    3. m

    Example \(\PageIndex{33}\)

    Simplify:

    1. \(5^{\frac{3}{5}}·5^{\frac{7}{5}}\)
    2. \(z^{\frac{1}{8}}·z^{\frac{7}{8}}\)
    3. \(n^{\frac{2}{7}}·n^{\frac{5}{7}}\).
    Answer
    1. 25
    2. z
    3. n

    We will use the Power Property in the next example.

    Example \(\PageIndex{34}\)

    Simplify:

    1. \((x^4)^{\frac{1}{2}}\)
    2. \((y^6)^{\frac{1}{3}}\)
    3. \((z^9)^{\frac{2}{3}}\).
    Answer
    1. \((x^4)^{\frac{1}{2}}\)
    To raise a power to a power, we multiply the exponents. \(x^{4·\frac{1}{2}}\)
    Simplify. \(x^2\)
    2. \((y^6)^{\frac{1}{3}}\)
    To raise a power to a power, we multiply the exponents. \(y^{6·\frac{1}{3}}\)
    Simplify. \(y^2\)
    3. \((z^9)^{\frac{2}{3}}\)
    To raise a power to a power, we multiply the exponents. \(z^{9·\frac{2}{3}}\)
    Simplify. \(z^6\)

    Example \(\PageIndex{35}\)

    Simplify:

    1. \((p^{10})^{\frac{1}{5}}\)
    2. \((q^8)^{\frac{3}{4}}\)
    3. \((x^6)^{\frac{4}{3}}\)
    Answer
    1. \(p^\)
    2. \(q^6\)
    3. \(x^8\)

    Example \(\PageIndex{36}\)

    Simplify:

    1. \((r^6)^{\frac{5}{3}}\)
    2. \((s^{12})^{\frac{3}{4}}\)
    3. \((m^9)^{\frac{2}{9}}\)
    Answer
    1. \(r^{10}\)
    2. \(s^9\)
    3. \(m^2\)

    The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

    Example \(\PageIndex{37}\)

    Simplify:

    1. \(\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}\)
    2. \(\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}\)
    3. \(\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}\).
    Answer
    1. \(\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}\)
    To divide with the same base, we subtract the exponents. \(x^{\frac{4}{3}−\frac{1}{3}}\)
    Simplify. x
    2. \(\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}\)
    To divide with the same base, we subtract the exponents. \(y^{\frac{3}{4}−\frac{1}{4}}\)
    Simplify. \(y^{\frac{1}{2}}\)
    3. \(\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}\)
    To divide with the same base, we subtract the exponents. \(z^{\frac{2}{3}−\frac{5}{3}}\)
    Rewrite without a negative exponent. \(\frac{1}{z}\)

    Example \(\PageIndex{38}\)

    Simplify:

    1. \(\frac{u^{\frac{5}{4}}}{u^{\frac{1}{4}}}\)
    2. \(\frac{v^{\frac{3}{5}}}{v^{\frac{2}{5}}}\)
    3. \(\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}\).
    Answer
    1. u
    2. \(v^{\frac{1}{5}}\)
    3. \(\frac{1}{x}\)

    Example \(\PageIndex{39}\)

    Simplify:

    1. \(\frac{c^{\frac{12}{5}}}{c^{\frac{2}{5}}}\)
    2. \(\frac{m^{\frac{5}{4}}}{m^{\frac{9}{4}}}\)
    3. \(\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}\).
    Answer
    1. \(c^2\)
    2. \(\frac{1}{m}\)
    3. \(\frac{1}{d}\)

    Sometimes we need to use more than one property. In the next two examples, we will use both the Product to a Power Property and then the Power Property.

    Example \(\PageIndex{40}\)

    Simplify:

    1. \((27u^{\frac{1}{2}})^{\frac{2}{3}}\)
    2. \((8v^{\frac{1}{4}})^{\frac{2}{3}}\).
    Answer
    1. \((27u^{\frac{1}{2}})^{\frac{2}{3}}\)
    First we use the Product to a Power Property. \((27)^{\frac{2}{3}}(u^{\frac{1}{2}})^{\frac{2}{3}}\)
    Rewrite 27 as a power of 3. \((3^3)^{\frac{2}{3}}(u^{\frac{1}{2}})^{\frac{2}{3}}\)
    To raise a power to a power, we multiply the exponents. \((3^2)(u^{\frac{1}{3}})\)
    Simplify. \(9u^{\frac{1}{3}}\)
    2. \((8v^{\frac{1}{4}})^{\frac{2}{3}}\).
    First we use the Product to a Power Property. \((8)^{\frac{2}{3}}(v^{\frac{1}{4}})^{\frac{2}{3}}\)
    Rewrite 8 as a power of 2. \((2^3)^{\frac{2}{3}}(v^{\frac{1}{4}})^{\frac{2}{3}}\)
    To raise a power to a power, we multiply the exponents. \((2^2)(v^{\frac{1}{6}})\)
    Simplify. \(4v^{\frac{1}{6}}\)

    Example \(\PageIndex{41}\)

    Simplify:

    1. \(32x^{\frac{1}{3}})^{\frac{3}{5}}\)
    2. \((64y^{\frac{2}{3}})^{\frac{1}{3}}\).
    Answer
    1. \(8x^{\frac{1}{5}}\)
    2. \(4y^{\frac{2}{9}}\)

    Example \(\PageIndex{42}\)

    Simplify:

    1. \((16m^{\frac{1}{3}})^{\frac{3}{2}}\)
    2. \((81n^{\frac{2}{5}})^{\frac{3}{2}}\).
    Answer
    1. \(64m^{\frac{1}{2}}\)
    2. \(729n^{\frac{3}{5}}\)

    Example \(\PageIndex{43}\)

    Simplify:

    1. \((m^{3}n^{9})^{\frac{1}{3}}\)
    2. \((p^{4}q^{8})^{\frac{1}{4}}\).
    Answer
    1. \((m^{3}n^{9})^{\frac{1}{3}}\)
    First we use the Product to a Power Property. \((m^{3})^{\frac{1}{3}}(n^{9})^{\frac{1}{3}}\)
    To raise a power to a power, we multiply the exponents. \(mn^3\)
    2. \((p^{4}q^{8})^{\frac{1}{4}}\)
    First we use the Product to a Power Property. \((p^{4})^{\frac{1}{4}}(q^{8})^{\frac{1}{4}}\)
    To raise a power to a power, we multiply the exponents. \(pq^2\)

    We will use both the Product and Quotient Properties in the next example.

    Exercise \(\PageIndex{44}\)

    Simplify:

    1. \(\frac{x^{\frac{3}{4}}·x^{−\frac{1}{4}}}{x^{−\frac{6}{4}}}\)
    2. \(\frac{y^{\frac{4}{3}}·y}{y^{−\frac{2}{3}}}\).
    Answer
    1. \(\frac{x^{\frac{3}{4}}·x^{−\frac{1}{4}}}{x^{−\frac{6}{4}}}\)
    Use the Product Property in the numerator, add the exponents. \(\frac{x^{\frac{2}{4}}}{x^{−\frac{6}{4}}}\)
    Use the Quotient Property, subtract the exponents. \(x^{\frac{8}{4}}\)
    Simplify. \(x^2\)
    2. \(\frac{y^{\frac{4}{3}}·y}{y^{−\frac{2}{3}}}\)
    Use the Product Property in the numerator, add the exponents. \(\frac{y^{\frac{7}{3}}}{y^{−\frac{2}{3}}}\)
    Use the Quotient Property, subtract the exponents. \(y^{\frac{9}{3}}\)
    Simplify. \(y^3\)

    Example \(\PageIndex{45}\)

    Simplify:

    1. \(\frac{m^{\frac{2}{3}}·m^{−\frac{1}{3}}}{m^{−\frac{5}{3}}}\)
    2. \(\frac{n^{\frac{1}{6}}·n}{n^{−\frac{11}{6}}}\).
    Answer
    1. \(m^2\)
    2. \(n^3\)

    Example \(\PageIndex{46}\)

    Simplify:

    1. \(\frac{u^{\frac{4}{5}}·u^{−\frac{2}{5}}}{u^{−\frac{13}{5}}}\)
    2. \(\frac{v^{\frac{1}{2}}·v}{v^{−\frac{7}{2}}}\).
    Answer
    1. \(u^3\)
    2. \(v^5\)

    Key Concepts

    • Summary of Exponent Properties
    • If a,b are real numbers and m,n are rational numbers, then
      • Product Property \(a^m·a^n=a^{m+n}\)
      • Power Property \((a^m)^n=a^{m·n}\)
      • Product to a Power \((ab)^m=a^{m}b^{m}\)
      • Quotient Property:

        \(\frac{a^m}{a^n}=a^{m−n} , a \ne 0, m>n\)

        \(\frac{a^m}{a^n}=\frac{1}{a^{n−m}}, a \ne 0, n>m\)

      • Zero Exponent Definition \(a^0=1, a \ne 0\)
      • Quotient to a Power Property \((\frac{a}{b})^m=\frac{a^m}{b^m}, b \ne 0\)

    Section Exercises

    Practice Makes Perfect

    Simplify Expressions with \(a^{\frac{1}{n}}\)

    In the following exercises, write as a radical expression.

    Example \(\PageIndex{47}\)

    1. \(x^{\frac{1}{2}}\)
    2. \(y^{\frac{1}{3}}\)
    3. \(z^{\frac{1}{4}}\)

    Example \(\PageIndex{48}\)

    1. \(r^{\frac{1}{2}}\)
    2. \(s^{\frac{1}{3}}\)
    3. \(t^{\frac{1}{4}}\)
    Answer
    1. \(\sqrt{r}\)
    2. \(\sqrt[3]{s}\)
    3. \(\sqrt[4]{t}\)

    Example \(\PageIndex{49}\)

    1. \(u^{\frac{1}{5}}\)
    2. \(v^{\frac{1}{9}}\)
    3. \(w^{\frac{1}{20}}\)

    Example \(\PageIndex{50}\)

    1. \(g^{\frac{1}{7}}\)
    2. \(h^{\frac{1}{5}}\)
    3. \(j^{\frac{1}{25}}\)
    Answer
    1. \(\sqrt[7]{g}\)
    2. \(\sqrt[5]{h}\)
    3. \(\sqrt[25]{j}\)
    In the following exercises, write with a rational exponent.

    Example \(\PageIndex{51}\)

    1. \(−\sqrt[7]{x}\)
    2. \(\sqrt[9]{y}\)
    3. \(\sqrt[5]{f}\)

    Example \(\PageIndex{52}\)

    1. \(\sqrt[8]{r}\)
    2. \(\sqrt[10]{s}\)
    3. \(\sqrt[4]{t}\)
    Answer
    1. \(r^{\frac{1}{8}}\)
    2. \(s^{\frac{1}{10}}\)
    3. \(t^{\frac{1}{4}}\)

    Example \(\PageIndex{53}\)

    1. \(\sqrt[3]{a}\
    2. \(\sqrt[12]{b}\)
    3. \(\sqrt{c}\)

    Example \(\PageIndex{54}\)

    1. \(\sqrt[5]{u}\)
    2. \(\sqrt{v}\)

    .

    Answer
    1. \(u^{\frac{1}{5}}\)
    2. \(v^{\frac{1}{2}}\)
    3. \(w^{\frac{1}{16}}\)

    Example \(\PageIndex{55}\)

    1. \(\sqrt[3]{7c}\)
    2. \(\sqrt[7]{12d}\)
    3. \(3\sqrt[4]{5f}\)

    Example \(\PageIndex{56}\)

    1. \(\sqrt[4]{5x}\)
    2. \(\sqrt[8]{9y}\)
    3. \(7\sqrt[5]{3z}\)
    Answer
    1. \((5x)^{\frac{1}{4}}\)
    2. \((9y)^{\frac{1}{8}}\)
    3. \(7(3z)^{\frac{1}{5}}\)

    Example \(\PageIndex{57}\)

    1. \(\sqrt{21p}\)
    2. \(\sqrt[4]{8q}\)
    3. \(\sqrt[6]{436r}\)

    Example \(\PageIndex{58}\)

    1. \(\sqrt[3]{25a}\)
    2. \(\sqrt{3b}\)

    .

    Answer
    1. \((25a)^{\frac{1}{3}}\)
    2. \((3b)^{\frac{1}{2}}\)
    3. \((40c)^{\frac{1}{10}}\)

    In the following exercises, simplify.

    Example \(\PageIndex{59}\)

    1. \(81^{\frac{1}{2}}\)
    2. \(125^{\frac{1}{3}}\)
    3. \(64^{\frac{1}{2}}\)

    Example \(\PageIndex{60}\)

    1. \(625^{\frac{1}{4}}\)
    2. \(243^{\frac{1}{5}}\)
    3. \(32^{\frac{1}{5}}\)
    Answer
    1. 5
    2. 3
    3. 2

    Example \(\PageIndex{61}\)

    1. \(16^{\frac{1}{4}}\)
    2. \(16^{\frac{1}{2}}\)
    3. \(3125^{\frac{1}{5}}\)

    Example \(\PageIndex{62}\)

    1. \(216^{\frac{1}{3}}\)
    2. \(32^{\frac{1}{5}}\)
    3. \(81^{\frac{1}{4}}\)
    Answer
    1. 6
    2. 2
    3. 3

    Example \(\PageIndex{63}\)

    1. \((−216)^{\frac{1}{3}}\)
    2. \(−216^{\frac{1}{3}}\)
    3. \((216)^{−\frac{1}{3}}\)

    Example \(\PageIndex{64}\)

    1. \((−243)^{\frac{1}{5}}\)
    2. \(−243^{\frac{1}{5}}\)
    3. \((243)^{−\frac{1}{5}}\)
    Answer
    1. −3
    2. −3
    3. \(\frac{1}{3}\)

    Example \(\PageIndex{65}\)

    1. \((−1)^{\frac{1}{3}}\)
    2. \(−1^{\frac{1}{3}}\)
    3. \((1)^{−\frac{1}{3}}\)

    Example \(\PageIndex{66}\)

    1. \((−1000)^{\frac{1}{3}}\)
    2. \(−1000^{\frac{1}{3}}\)
    3. \((1000)^{−\frac{1}{3}}\)
    Answer
    1. −10
    2. −10
    3. \(\frac{1}{10}

    Example \(\PageIndex{67}\)

    1. \((−81)^{\frac{1}{4}}\)
    2. \(−81^{\frac{1}{4}}\)
    3. \((81)^{−\frac{1}{4}}\)

    Example \(\PageIndex{68}\)

    1. \((−49)^{\frac{1}{2}}\)
    2. \(−49^{\frac{1}{2}}\)
    3. \((49)^{−\frac{1}{2}}\)
    Answer
    1. not a real number
    2. −7
    3. \(\frac{1}{7}\)

    Example \(\PageIndex{69}\)

    1. \((−36)^{\frac{1}{2}}\)
    2. \(−36^{\frac{1}{2}}\)
    3. \((36)^{−\frac{1}{2}}\)

    Example \(\PageIndex{70}\)

    1. \((−1)^{\frac{1}{4}}\)
    2. \((1)^{−\frac{1}{4}}\)
    3. \(−1^{\frac{1}{4}}\)
    Answer
    1. not a real number
    2. 1
    3. −1

    Example \(\PageIndex{71}\)

    1. \((−100)^{\frac{1}{2}}\)
    2. \(−100^{\frac{1}{2}}\)
    3. \((100)^{−\frac{1}{2}}\)

    Example \(\PageIndex{72}\)

    1. \((−32)^{\frac{1}{5}}\)
    2. \((243)^{−\frac{1}{5}}\)
    3. \(−125^{\frac{1}{3}}\)
    Answer
    1. −2
    2. \(\frac{1}{3}\)
    3. −5

    Simplify Expressions with \(a^{\frac{m}{n}}\)

    In the following exercises, write with a rational exponent.

    Example \(\PageIndex{73}\)

    1. \(\sqrt{m^5}\)
    2. \(\sqrt[3]{n^2}\)
    3. \(\sqrt[4]{p^3}\)

    Example \(\PageIndex{74}\)

    1. \(\sqrt[4]{r^7}\)
    2. \(\sqrt[5]{s^3}\)
    3. \(\sqrt[3]{t^7}\)
    Answer
    1. \(r^{\frac{7}{4}}\)
    2. \(s^{\frac{3}{5}}\)
    3. \(t^{\frac{7}{3}}\)

    Exercise \(\PageIndex{75}\)

    1. \(\sqrt[5]{u^2}\)
    2. \(\sqrt[5]{v^8}\)
    3. \(\sqrt[9]{w^4}\)

    Example \(\PageIndex{76}\)

    1. \(\sqrt[3]{a}\)
    2. \(\sqrt[5]{b}\)
    3. \(\sqrt[3]{c^5}\)
    Answer
    1. \(a^{\frac{1}{3}}\)
    2. \(b^{\frac{1}{5}}\)
    3. \(c^{\frac{5}{3}}\)
    In the following exercises, simplify.

    Example \(\PageIndex{77}\)

    1. \(16^{\frac{3}{2}}\)
    2. \(8^{\frac{2}{3}}\)
    3. \(10,000^{\frac{3}{4}}\)

    Example \(\PageIndex{78}\)

    1. \(1000^{\frac{2}{3}}\)
    2. \(25^{\frac{3}{2}}\)
    3. \(32^{\frac{3}{5}}\)
    Answer
    1. 100
    2. 125
    3. 8

    Example \(\PageIndex{79}\)

    1. \(275^{\frac{3}{2}}\)
    2. \(16^{\frac{5}{4}}\)
    3. \(32^{\frac{2}{5}}\)

    Example \(\PageIndex{80}\)

    1. \(16^{\frac{3}{2}}\)
    2. \(125^{\frac{5}{3}}\)
    3. \(64^{\frac{4}{3}}\)
    Answer
    1. 64
    2. 3125
    3. 256

    Example \(\PageIndex{81}\)

    1. \(32^{\frac{2}{5}}\)
    2. \(27^{−\frac{2}{3}}\)
    3. \(25^{−\frac{3}{2}}\)

    Example \(\PageIndex{82}\)

    1. \(64^{\frac{5}{2}}\)
    2. \(81^{−\frac{3}{2}}\)
    3. \(27^{−\frac{4}{3}}\)
    Answer
    1. 32,768
    2. \(\frac{1}{729}\)
    3. \(\frac{1}{81}\)

    Example \(\PageIndex{83}\)

    1. \(25^{\frac{3}{2}}\)
    2. \(9^{−\frac{3}{2}}\)
    3. \((−64)^{\frac{2}{3}}\)

    Example \(\PageIndex{84}\)

    1. \(100^{\frac{3}{2}}\)
    2. \(49^{−\frac{5}{2}}\)
    3. \((−100)^{\frac{3}{2}}\)
    Answer
    1. 1000
    2. \(\frac{1}{16,807}\)
    3. not a real numbe

    Example \(\PageIndex{85}\)

    1. \(−9^{\frac{3}{2}}\)
    2. \(−9^{−\frac{3}{2}}\)
    3. \((−9)^{\frac{3}{2}}\)

    Example \(\PageIndex{86}\)

    1. \(−64^{\frac{3}{2}}\)
    2. \(−64^{−\frac{3}{2}}\)
    3. \((−64)^{\frac{3}{2}}\)
    Answer
    1. −512
    2. \(−\frac{1}{512}\)
    3. not a real number

    Example \(\PageIndex{87}\)

    1. \(−100^{\frac{3}{2}}\)
    2. \(−100^{−\frac{3}{2}}\)
    3. \((−100)^{\frac{3}{2}}\)

    Example \(\PageIndex{88}\)

    1. \(−49^{\frac{3}{2}}\)
    2. \(−49^{−\frac{3}{2}}\)
    3. \((−49)^{\frac{3}{2}}\)
    Answer
    1. −343
    2. \(−\frac{1}{343}\)
    3. not a real number

    Use the Laws of Exponents to Simplify Expressions with Rational Exponents

    In the following exercises, simplify.

    Example \(\PageIndex{89}\)

    1. \(4^{\frac{5}{8}}·4^{\frac{11}{8}}\)
    2. \(m^{\frac{7}{12}}·m^{\frac{17}{12}}\)
    3. \(p^{\frac{3}{7}}·p^{\frac{18}{7}}\)

    Example \(\PageIndex{90}\)

    1. \(6^{\frac{5}{2}}·6^{\frac{12}{6}}\)
    2. \(n^{\frac{2}{10}}·n^{\frac{8}{10}}\)
    3. \(q^{\frac{2}{5}}·q^{\frac{13}{5}}\)
    Answer
    1. 216
    2. n
    3. \(q^3\)

    Example \(\PageIndex{91}\)

    1. \(5^{\frac{1}{2}}·5^{\frac{7}{2}}\)
    2. \(c^{\frac{3}{4}}·c^{\frac{9}{4}}\)
    3. \(d^{\frac{3}{5}}·d^{\frac{2}{5}}\)

    Example \(\PageIndex{92}\)

    1. \(10^{\frac{1}{3}}·10^{\frac{5}{3}}\)
    2. \(x^{\frac{5}{6}}·x^{\frac{7}{6}}\)
    3. \(y^{\frac{11}{8}}·y^{\frac{21}{8}}\)
    Answer
    1. 100
    2. \(x^2\)
    3. \(y^4\)

    Example \(\PageIndex{93}\)

    1. \((m^6)^{\frac{5}{2}}\)
    2. \((n^9)^{\frac{4}{3}}\)
    3. \((p^{12})^{\frac{3}{4}}\)

    Example \(\PageIndex{94}\)

    1. \((a^{12})^{\frac{1}{6}}\)
    2. \((b^{15})^{\frac{3}{5}}\)
    3. \((c^{11})^{\frac{1}{11}}\)
    Answer
    1. \(a^2\)
    2. \(b^9\)
    3. c

    Example \(\PageIndex{95}\)

    1. \((x^{12})^{\frac{2}{3}}\)
    2. \((y^{20})^{\frac{2}{5}}\)
    3. \((z^{16})^{\frac{1}{16}}\)

    Example \(\PageIndex{96}\)

    1. \((h^6)^{\frac{4}{3}}\)
    2. \((k^{12})^{\frac{3}{4}}\)
    3. \((j^{10})^{\frac{7}{5}}\)
    Answer
    1. \(h^8\)
    2. \(k^9\)
    3. \(j^{14}\)

    Example \(\PageIndex{97}\)

    1. \(\frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}\)
    2. \(\frac{y^{\frac{5}{2}}}{y^{\frac{1}{2}}}\)
    3. \(\frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}\)

    Example \(\PageIndex{98}\)

    1. \(\frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}}\)
    2. \(\frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}}\)
    3. \(\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}\)
    Answer
    1. s
    2. \(z^2\)
    3. \(\frac{1}{w}\)

    Example \(\PageIndex{99}\)

    1. \(\frac{t^{\frac{12}{5}}}{t^{\frac{7}{5}}}\)
    2. \(\frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}}\)
    3. \(\frac{m^{\frac{13}{8}}}{m^{\frac{5}{8}}}\)

    Example \(\PageIndex{100}\)

    1. \(\frac{u^{\frac{13}{9}}}{u^{\frac{4}{9}}}\)
    2. \(\frac{r^{\frac{15}{7}}}{r^{\frac{8}{7}}}\)
    3. \(\frac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}\)
    Answer
    1. u
    2. r
    3. \(\frac{1}{n}\)

    Example \(\PageIndex{101}\)

    1. \((9p^{\frac{2}{3}})^{\frac{5}{2}}\)
    2. \((27q^{\frac{3}{2}})^{\frac{4}{3}}\)​​​​​

    Example \(\PageIndex{102}\)

    1. \((81r^{\frac{4}{5}})^{\frac{1}{4}}\)
    2. \((64s^{\frac{3}{7}})^{\frac{1}{6}}\)
    Answer
    1. \(3r^{\frac{1}{5}}\)
    2. \(2s^{\frac{1}{14}}\)

    Example \(\PageIndex{103}\)

    1. \((16u^{\frac{1}{3}})^{\frac{3}{4}}\)
    2. \((100v^{\frac{2}{5}})^{\frac{3}{2}}\)

    Example \(\PageIndex{104}\)

    1. \((27m^{\frac{3}{4}})^{\frac{2}{3}}\)
    2. \((625n^{\frac{8}{3}})^{\frac{3}{4}}\)
    Answer
    1. \(9m^{\frac{1}{2}}\)
    2. \(125n^2\)

    Example \(\PageIndex{105}\)

    1. \((x^{8}y^{10})^{\frac{1}{2}}\)
    2. \((a^{9}by ^{12})^{\frac{1}{3}}\)

    Example \(\PageIndex{106}\)

    1. \((r^{8}s^{4})^{\frac{1}{4}}\)
    2. \((u^{15}v^{20})^{\frac{1}{5}}\)
    Answer
    1. \(r^{2}s\)
    2. \(u^{3}v^{4}\)

    Example \(\PageIndex{107}\)

    1. \((a^{6}b^{16})^{\frac{1}{2}}\)
    2. \((j^{9}k^{6})^{\frac{2}{3}}\)

    Example \(\PageIndex{108}\)

    1. \((r^{16}s^{10})^{\frac{1}{2}}\)
    2. \((u^{10}v^5)^{\frac{4}{5}}\)
    Answer
    1. \(r^{8}s^{5}\)
    2. \(u^{8}v^{4}\)

    Example \(\PageIndex{109}\)

    1. \(\frac{r^{\frac{5}{2}}·r^{−\frac{1}{2}}}{r^{−\frac{3}{2}}}\)
    2. \(\frac{s^{\frac{1}{5}}·s}{s^{−\frac{9}{5}}}\)

    Example \(\PageIndex{110}\)

    1. \(\frac{a^{\frac{3}{4}}·a^{−\frac{1}{4}}}{a^{−\frac{10}{4}}}\)
    2. \(\frac{b^{\frac{2}{3}}·b}{b^{−\frac{7}{3}}}\)
    Answer
    1. \(a^3\)
    2. \(b^4\)

    Example \(\PageIndex{111}\)

    1. \(\frac{c^{\frac{5}{3}}·c^{−\frac{1}{3}}}{c^{−\frac{2}{3}}}\)
    2. \(\frac{d^{\frac{3}{5}}·d}{d^{−\frac{2}{5}}}\)

    Example \(\PageIndex{112}\)

    1. \(\frac{m^{\frac{7}{4}}·m^{−\frac{5}{4}}}{m^{−\frac{2}{4}}}\)
    2. \(\frac{n^{\frac{3}{7}}·n}{n^{−\frac{4}{7}}}\)
    Answer
    1. m
    2. \(n^2\)

    Example \(\PageIndex{113}\)

    \(4^{\frac{5}{2}}·4^{\frac{1}{2}}\)

    Example \(\PageIndex{114}\)

    \(n^{\frac{2}{6}}·n^{\frac{4}{6}}\)

    Answer

    n

    Example \(\PageIndex{115}\)

    \((a^24)^ {\frac{1}{6}}\)

    Example \(\PageIndex{116}\)

    \((b^{10})^{\frac{3}{5}}\)

    Answer

    \(b^6\)

    Example \(\PageIndex{117}\)

    \(\frac{w^{\frac{2}{5}}}{w^{\frac{7}{5}}}\)

    Example \(\PageIndex{118}\)

    \(\frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}}\)

    Answer

    \(\frac{1}{z^2}\)

    Example \(\PageIndex{119}\)

    \((27r^{\frac{3}{5}})^{\frac{1}{3}}\)

    Example \(\PageIndex{120}\)

    \((64s^{\frac{3}{5}})^{\frac{1}{6}}\)

    Answer

    \(2s^{\frac{1}{10}}\)

    Example \(\PageIndex{121}\)

    \((r^{9}s^{12})^{\frac{1}{3}}\)

    Example \(\PageIndex{122}\)

    \((u^{12}v^{18})^{\frac{1}{6}}\)

    Answer

    \(u^{2}v^{3}\)​​​​​​

    Everyday Math

    Example \(\PageIndex{123}\)

    Landscaping Joe wants to have a square garden plot in his backyard. He has enough compost to cover an area of 144 square feet. Simplify \(144^{\frac{1}{2}}\) to find the length of each side of his garden.

    Example \(\PageIndex{124}\)

    Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify \(242^{\frac{1}{2}}\) to find the length of each side of his patio.Round to the nearest tenth of a foot.

    Answer

    15.6 feet

    Example \(\PageIndex{125}\)

    Gravity While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify \(\frac{12^{\frac{1}{2}}}{16^{\frac{1}{2}}}\) to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

    Example \(\PageIndex{126}\)

    Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify \(\frac{1024^{\frac{1}{2}}}{16^{\frac{1}{2}}}\) to find how many seconds it took for the flare to reach the water.

    Answer

    8 seconds

    Writing Exercises

    Example \(\PageIndex{127}\)

    Show two different algebraic methods to simplify \(4^{\frac{3}{2}}\). Explain all your steps.

    Example \(\PageIndex{128}\)

    Explain why the expression \((−16)^{\frac{3}{2}}\) cannot be evaluated.

    Chapter 9 Review Exercises

    Simplify and Use Square Roots

    Simplify Expressions with Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{1}\)

    \(\sqrt{64}\)

    Exercise \(\PageIndex{2}\)

    \(\sqrt{144}\)

    Answer

    12

    Exercise \(\PageIndex{3}\)

    \(−\sqrt{25}\)

    Exercise \(\PageIndex{4}\)

    \(−\sqrt{81}\)

    Answer

    −9

    Exercise \(\PageIndex{5}\)

    \(\sqrt{−9}\)

    Exercise \(\PageIndex{6}\)

    \(\sqrt{−36}\)

    Answer

    not a real number

    Exercise \(\PageIndex{7}\)

    \(\sqrt{64}+\sqrt{225}\)

    Exercise \(\PageIndex{8}\)

    \(\sqrt{64+225}\)

    Answer

    17

    Estimate Square Roots

    In the following exercises, estimate each square root between two consecutive whole numbers.

    Exercise \(\PageIndex{9}\)

    \(\sqrt{28}\)

    Exercise \(\PageIndex{10}\)

    \(\sqrt{155}\)

    Answer

    \(12<\sqrt{155}<13\)

    Approximate Square Roots

    In the following exercises, approximate each square root and round to two decimal places.

    Exercise \(\PageIndex{11}\)

    \(\sqrt{15}\)

    Example \(\PageIndex{12}\)

    \(\sqrt{57}\)

    Answer

    7.55

    Simplify Variable Expressions with Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{13}\)

    \(\sqrt{q^2}\)

    Exercise \(\PageIndex{14}\)

    \(\sqrt{64b^2}\)

    Answer

    8b

    Exercis\e \(\PageIndex{15}\)

    \(−\sqrt{121a^2}\)

    Exercise \(\PageIndex{16}\)

    \(\sqrt{225m^{2}n^{2}}\)

    Answer

    15mn

    Exercise \(\PageIndex{17}\)

    \(−\sqrt{100q^2}\)

    Exercise \(\PageIndex{18}\)

    \(\sqrt{49y^2}\)

    Answer

    7y

    Exercise \(\PageIndex{19}\)

    \(\sqrt{4a^{2}b^{2}}\)

    Exercise \(\PageIndex{20}\)

    \(\sqrt{121c^{2}d^{2}}\)

    Answer

    11cd

    Simplify Square Roots

    Use the Product Property to Simplify Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{21}\)

    \(\sqrt{300}\)

    Exercise \(\PageIndex{22}\)

    \(\sqrt{98}\)

    Answer

    \(7\sqrt{2}\)

    Exercise \(\PageIndex{23}\)

    \(\sqrt{x^{13}}\)

    Exercise \(\PageIndex{24}\)

    \(\sqrt{y^{19}}\)

    Answer

    \(y^{9}\sqrt{y}\)

    Exercise \(\PageIndex{25}\)

    \(\sqrt{16m^4}\)

    Exercise \(\PageIndex{26}\)

    \(\sqrt{36n^{13}}\)

    Answer

    \(6n^{6}\sqrt{n}\)

    Exercise \(\PageIndex{27}\)

    \(\sqrt{288m^{21}}\)

    Exercise \(\PageIndex{28}\)

    \(\sqrt{150n^7}\)

    Answer

    \(5n^3\sqrt{6n}\)

    Exercise \(\PageIndex{29}\)

    \(\sqrt{48r^{5}s^{4}}\)

    Exercise \(\PageIndex{30}\)

    \(\sqrt{108r^{5}s^{3}}\)

    Answer

    \(6r^{2}s\sqrt{3rs}\)

    Exercise \(\PageIndex{31}\)

    \(\frac{10−\sqrt{50}}{5}\)

    Exercise \(\PageIndex{32}\)

    \(\frac{6+\sqrt{72}}{6}\)

    Answer

    \(1+\sqrt{2}\)

    Use the Quotient Property to Simplify Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{33}\)

    \(\sqrt{\frac{16}{25}}\)

    Exercise \(\PageIndex{34}\)

    \(\sqrt{\frac{81}{36}}\)

    Answer

    \(\frac{3}{2}\)

    Exercise \(\PageIndex{35}\)

    \(\sqrt{\frac{x^8}{x^4}}\)

    Exercise \(\PageIndex{36}\)

    \(\sqrt{\frac{y^6}{y^2}}\)

    Answer

    \(y^2\)

    Exercise \(\PageIndex{37}\)

    \(\sqrt{\frac{98p^6}{2p^2}}\)

    Exercise \(\PageIndex{38}\)

    \(\sqrt{\frac{72q^8}{2q^4}}\)

    Answer

    \(6q^2\)

    Exercise \(\PageIndex{39}\)

    \(\sqrt{\frac{65}{121}}\)

    Exercise \(\PageIndex{40}\)

    \(\sqrt{\frac{26}{169}}\)

    Answer

    \(\frac{\sqrt{26}}{13}\)

    Exercise \(\PageIndex{41}\)

    \(\sqrt{\frac{64x^4}{25x^2}}\)

    Exercise \(\PageIndex{42}\)

    \(\sqrt{\frac{36r^{10}}{16r^5}}\)

    Answer

    \(\frac{3r^2\sqrt{r}}{2}\)

    Exercise \(\PageIndex{43}\)

    \(\sqrt{\frac{48p^{3}q^{5}}{27pq}}\)

    Exercise \(\PageIndex{44}\)

    \(\sqrt{\frac{12r^{5}s^{7}}{75r^{2}s}}\)

    Answer

    \(\frac{2rs^3\sqrt{r}}{5}\)

    Add and Subtract Square Roots

    Add and Subtract Like Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{45}\)

    \(3\sqrt{2}+\sqrt{2}\)

    Exercise \(\PageIndex{46}\)

    \(5\sqrt{5}+7\sqrt{5}\)

    Answer

    \(12\sqrt{5}\)

    Exercise \(\PageIndex{47}\)

    \(4\sqrt{y}+4\sqrt{y}\)

    Exercise \(\PageIndex{48}\)

    \(6\sqrt{m}−2\sqrt{m}\)

    Answer

    \(4\sqrt{m}\)

    Exercise \(\PageIndex{49}\)

    \(−3\sqrt{7}+2\sqrt{7}−\sqrt{7}\)

    Exercise \(\PageIndex{50}\)

    \(8\sqrt{13}+2\sqrt{3}+3\sqrt{13}\)

    Answer

    \(11\sqrt{13}+2\sqrt{3}\)

    Exercise \(\PageIndex{51}\)

    \(3\sqrt{5xy}−\sqrt{5xy}+3\sqrt{5xy}\)

    Exercise \(\PageIndex{52}\)

    \(2\sqrt{3rs}+\sqrt{3rs}−5\sqrt{rs}\)

    Answer

    \(3\sqrt{3rs}−5\sqrt{rs}\)

    Add and Subtract Square Roots that Need Simplification

    In the following exercises, simplify.

    Exercise \(\PageIndex{53}\)

    \(\sqrt{32}+3\sqrt{2}\)

    Exercise \(\PageIndex{54}\)

    \(\sqrt{8}+\sqrt{32}\)

    Answer

    \(5\sqrt{2}\)

    Exercise \(\PageIndex{55}\)

    \(\sqrt{72}+\sqrt{50}\)

    Exercise \(\PageIndex{56}\)

    \(\sqrt{48}+\sqrt{75}\)

    Answer

    \(9\sqrt{3}\)

    Exercise \(\PageIndex{57}\)

    \(3\sqrt{32}+\sqrt{98}\)

    Exercise \(\PageIndex{58}\)

    \(\frac{1}{3}\sqrt{27}−\frac{1}{8}\sqrt{192}\)

    Answer

    0

    Exercise \(\PageIndex{59}\)

    \(\sqrt{50y^5}−\sqrt{72y^5}\)

    Exercise \(\PageIndex{60}\)

    Add exercises text here.

    Answer

    \(17n^2\sqrt{2}\)

    Multiply Square Roots

    Multiply Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{61}\)

    \(\sqrt{2}·\sqrt{20}\)

    Exercise \(\PageIndex{62}\)

    \(2\sqrt{2}·6\sqrt{14}\)

    Answer

    \(24\sqrt{7}\)

    Exercise \(\PageIndex{63}\)

    \(\sqrt{2m^2}·\sqrt{20m^4}\)

    Exercise \(\PageIndex{64}\)

    \((\sqrt{62y})(\sqrt{350y^3})\)

    Answer

    \(180y^2\)

    Exercise \(\PageIndex{65}\)

    \((6\sqrt{3v^4})(5\sqrt{30v})\)

    Exercise \(\PageIndex{66}\)

    \((\sqrt{8})^2\)

    Answer

    8

    Exercise \(\PageIndex{67}\)

    \((−\sqrt{10})^2\)

    Exercise \(\PageIndex{68}\)

    \((2\sqrt{5})(5\sqrt{5})\)

    Answer

    50

    Exercise \(\PageIndex{69}\)

    \((−3\sqrt{3})(5\sqrt{18})\)

    ​​​​​​​Use Polynomial Multiplication to Multiply Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{70}\)

    \(10(2−\sqrt{7})\)

    Answer

    \(20−10\sqrt{7}\)

    Exercise \(\PageIndex{71}\)

    \(\sqrt{3}(4+\sqrt{12})\)

    Exercise \(\PageIndex{72}\)

    \((5+\sqrt{2})(3−\sqrt{2})\)

    Answer

    \(13−2\sqrt{2}\)

    Exercise \(\PageIndex{73}\)

    \((5−3\sqrt{7})(1−2\sqrt{7})\)

    Exercise \(\PageIndex{74}\)

    \((1−3\sqrt{x})(5+2\sqrt{x})\)

    Answer

    \(5−13\sqrt{x}−6x\)

    Exercise \(\PageIndex{75}\)

    \((3+4\sqrt{y})(10−\sqrt{y})\)

    Exercise \(\PageIndex{76}\)

    \((1+6\sqrt{p})^2\)

    Answer

    \(1+12\sqrt{p}+36p\)

    Exercise \(\PageIndex{77}\)

    \((2−6\sqrt{5})^2\)

    Exercise \(\PageIndex{78}\)

    \((3+2\sqrt{7})(3−2\sqrt{7})\)

    Answer

    −19

    Exercise \(\PageIndex{79}\)

    \((6−\sqrt{11})(6+\sqrt{11})\)

    Divide Square Roots

    Divide Square Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{80}\)

    \(\frac{\sqrt{75}}{10}\)

    Answer

    \(\frac{\sqrt{3}}{2}\)

    Exercise \(\PageIndex{81}\)

    \(\frac{2−\sqrt{12}}{6}\)

    Exercise \(\PageIndex{82}\)

    \(\frac{\sqrt{48}}{\sqrt{27}}\)

    Answer

    \(\frac{4}{3}\)

    Exercise \(\PageIndex{83}\)

    \(\frac{\sqrt{75x^7}}{\sqrt{3x^3}}\)

    Exercise \(\PageIndex{84}\)

    \(\frac{\sqrt{20y^5}}{\sqrt{2y}}\)

    Answer

    \(y^2\sqrt{10}\)

    Exercise \(\PageIndex{85}\)

    \(\frac{\sqrt{98p^{6}q^{4}}}{\sqrt{2p^{4}q^{8}}}\)

    ​​​​​​​Rationalize a One Term Denominator

    In the following exercises, rationalize the denominator.

    Exercise \(\PageIndex{86}\)

    \(\frac{10}{\sqrt{15}}\)

    Answer

    \(\frac{2\sqrt{15}}{3}\)

    Exercise \(\PageIndex{87}\)

    \(\frac{6}{\sqrt{6}}\)

    Exercise \(\PageIndex{88}\)

    \(\frac{5}{3\sqrt{5}}\)

    Answer

    \(\frac{\sqrt{5}}{3}\)

    Exercise \(\PageIndex{89}\)

    \(\frac{10}{2\sqrt{6}}\)

    Exercise \(\PageIndex{90}\)

    \(\sqrt{\frac{3}{28}}\)

    Answer

    \(\frac{\sqrt{21}}{14}\)

    Exercise \(\PageIndex{91}\)

    \(\sqrt{\frac{9}{75}}\)

    ​​​​​​​Rationalize a Two Term Denominator

    In the following exercises, rationalize the denominator.

    Exercise \(\PageIndex{92}\)

    \(\frac{4}{4+\sqrt{27}}\)

    Answer

    \(\frac{16−12\sqrt{3}}{−11}\)

    Exercise \(\PageIndex{93}\)

    \(\frac{5}{2−\sqrt{10}}\)

    Exercise \(\PageIndex{94}\)

    \(\frac{4}{2−\sqrt{5}}\)

    Answer

    \(−8−4\sqrt{5}\)

    Exercise \(\PageIndex{95}\)

    \(\frac{5}{4−\sqrt{8}}\)

    Exercise \(\PageIndex{96}\)

    \(\frac{\sqrt{2}}{\sqrt{p}+\sqrt{3}}\)

    Answer

    \(\frac{\sqrt{2p}−\sqrt{6}}{p−3}\)

    Exercise \(\PageIndex{97}\)

    \(\frac{\sqrt{x}−\sqrt{2}}{\sqrt{x}+\sqrt{2}}\)

    Solve Equations with Square Roots

    Solve Radical Equations

    In the following exercises, solve the equation.

    Exercise \(\PageIndex{98}\)

    \(\sqrt{7z+1}=6\)

    Answer

    5

    Exercise \(\PageIndex{99}\)

    \(\sqrt{4u−2}−4=0\)

    Exercise \(\PageIndex{100}\)

    \(\sqrt{6m+4}−5=0\)

    Answer

    \(\frac{7}{2}\)

    Exercise \(\PageIndex{101}\)

    \(\sqrt{2u−3}+2=0\)

    Exercise \(\PageIndex{102}\)

    \(\sqrt{u−4}+4=u\)

    Answer

    no solution

    Exercise \(\PageIndex{103}\)

    \(\sqrt{v−9}+9=0\)​​​​​​​

    Exercise \(\PageIndex{104}\)

    \(\sqrt{r−4}−r=−10\)

    Answer

    13

    Exercise \(\PageIndex{105}\)

    \(\sqrt{s−9}−s=−9\)

    Exercise \(\PageIndex{106}\)

    \(2\sqrt{2x−7}−4=8\)

    Answer

    \(\frac{43}{2}\)

    Exercise \(\PageIndex{107}\)

    \(\sqrt{2−x}=\sqrt{2x−7}\)

    Exercise \(\PageIndex{108}\)

    \(\sqrt{a}+3=\sqrt{a+9}\)

    Answer

    0

    Exercise \(\PageIndex{109}\)

    \(\sqrt{r}+3=\sqrt{r+4}\)​​​​​​​

    Exercise \(\PageIndex{110}\)

    \(\sqrt{u}+2=\sqrt{u+5}\)

    Answer

    \(\frac{11}{6}\)

    Exercise \(\PageIndex{111}\)

    \(\sqrt{n+11}−1=\sqrt{n+4}\)

    Exercise \(\PageIndex{112}\)

    \(\sqrt{y+5}+1=\sqrt{2y+3}\)

    Answer

    11

    ​​​​​​​Use Square Roots in Applications

    In the following exercises, solve. Round approximations to one decimal place.

    Exercise \(\PageIndex{113}\)

    A pallet of sod will cover an area of about 600 square feet. Trinh wants to order a pallet of sod to make a square lawn in his backyard. Use the formula \(s=\sqrt{A}\) to find the length of each side of his lawn.​​​​​​​

    Exercise \(\PageIndex{114}\)

    A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the package to reach the hiker.

    Answer

    7.5 seconds

    Exercise \(\PageIndex{115}\)

    Officer Morales measured the skid marks of one of the cars involved in an accident. The length of the skid marks was 245 feet. Use the formula \(s=\sqrt{24d}\) to find the speed of the car before the brakes were applied.

    Higher Roots

    Simplify Expressions with Higher Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{116}\)

    1. \(\sqrt[6]{64}\)
    2. \(\sqrt[3]{64}\)
    Answer
    1. 2
    2. 4

    Exercise \(\PageIndex{117}\)

    1. \(\sqrt[3]{−27}\)
    2. \(\sqrt[4]{−64}\)

    Exercise \(\PageIndex{118}\)

    1. \(\sqrt[9]{d^9}\)
    2. \(\sqrt[8]{v^8}\)
    Answer
    1. d
    2. |v|

    Exercise \(\PageIndex{119}\)

    1. \(\sqrt[5]{a^{10}}\)
    2. \(\sqrt[3]{b^{27}}\)

    Exercise \(\PageIndex{120}\)

    1. \(\sqrt[4]{16x^8}\)
    2. \(\sqrt[6]{64y^{12}}\)
    Answer
    1. \(2x^2\)
    2. \(2y^2\)

    Exercise \(\PageIndex{121}\)

    1. \(\sqrt[7]{128r^{14}}\)
    2. \(\sqrt[4]{81s^{24}}\)

    Use the Product Property to Simplify Expressions with Higher Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{122}\)

    1. \(\sqrt[9]{d^9}\)

    .

    Answer
    1. d

    .​​​​​​​​​​​​​​

    Exercise \(\PageIndex{123}\)

    1. \(\sqrt[3]{54}\)
    2. \(\sqrt[4]{128}\)

    Exercise \(\PageIndex{124}\)

    1. \(\sqrt[5]{64c^8}\)
    2. \(\sqrt[4]{48d^7}\)
    Answer
    1. \(2c\sqrt[5]{2c^3}\)
    2. \(2d\sqrt[4]{3d^3}\)

    Exercise \(\PageIndex{125}\)

    1. \(\sqrt[3]{343q^7}\)
    2. \(\sqrt[6]{192r^9}\)

    Exercise \(\PageIndex{126}\)

    1. \(\sqrt[3]{−500}\)
    2. \(\sqrt[4]{−16}\)
    Answer
    1. \(−5\sqrt[3]{4}\)
    2. not a real number
    ​​​​​​​Use the Quotient Property to Simplify Expressions with Higher Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{127}\)

    \(\sqrt[5]{\frac{r^{10}}{r^5}}\)

    Exercise \(\PageIndex{128}\)

    \(\sqrt[3]{\frac{w^{12}}{w^2}}\)

    Answer

    \(w^3\sqrt[3]{w}\)

    Exercise \(\PageIndex{129}\)

    \(\sqrt[4]{\frac{64y^8}{4y^5}}\)

    Exercise \(\PageIndex{130}\)

    \(\sqrt[3]{\frac{54z^9}{2z^3}}\)

    Answer

    \(3z^2\)

    Exercise \(\PageIndex{131}\)

    \(\sqrt[6]{\frac{64a^7}{b^2}}\)

    ​​​​​​​Add and Subtract Higher Roots

    In the following exercises, simplify.

    Exercise \(\PageIndex{132}\)

    \(4\sqrt[5]{20}−2\sqrt[5]{20}\)

    Answer

    \(2\sqrt[5]{20}\)

    Exercise \(\PageIndex{133}\)

    \(4\sqrt[3]{18}+3\sqrt[3]{18}\)

    Exercise \(\PageIndex{134}\)

    \(\sqrt[4]{1250}−\sqrt[4]{162}\)

    Answer

    \(2\sqrt[4]{2}\)

    Exercise \(\PageIndex{135}\)

    \(\sqrt[3]{640c^5}−\sqrt[3]{−80c^3}\)

    Exercise \(\PageIndex{136}\)

    \(\sqrt[5]{96t^8}+\sqrt[5]{486t^4}\)

    Answer

    \(2t^\sqrt[5]{3t^3}+3\sqrt[5]{2t^4}\)

    Rational Exponents

    Simplify Expressions with \(a^{\frac{1}{n}}\)

    In the following exercises, write as a radical expression.

    Exercise \(\PageIndex{137}\)

    \(r^{\frac{1}{8}}\)

    Exercise \(\PageIndex{138}\)

    \(s^{\frac{1}{10}}\)

    Answer

    .

    ​​​​​​​In the following exercises, write with a rational exponent.

    Exercise \(\PageIndex{139}\)

    \(\sqrt[5]{u}\)

    Exercise \(\PageIndex{140}\)

    \(\sqrt[6]{v}\)

    Answer

    \(v^{\frac{1}{6}}\)

    Exercise \(\PageIndex{141}\)

    \(\sqrt[3]{9m}\)

    Exercise \(\PageIndex{142}\)

    \(\sqrt[6]{10z}\)

    Answer

    \((10z)^{\frac{1}{6}}\)

    ​​​​​​​In the following exercises, simplify.

    Exercise \(\PageIndex{143}\)

    \(16^{\frac{1}{4}}\)

    Exercise \(\PageIndex{144}\)

    \(32^{\frac{1}{5}}\)

    Answer

    2

    Exercise \(\PageIndex{145}\)

    \((−125)^{\frac{1}{3}}\)

    Exercise \(\PageIndex{146}\)

    \((125)^{−\frac{1}{3}}\)

    Answer

    \(\frac{1}{5}\)

    Exercise \(\PageIndex{147}\)

    \((−9)^{\frac{1}{2}}\)

    Exercise \(\PageIndex{148}\)

    \((36)^{−\frac{1}{2}}\)

    Answer

    \(\frac{1}{6}\)

    ​​​​​​​Simplify Expressions with \(a^{\frac{m}{n}}\)

    In the following exercises, write with a rational exponent.

    Exercise \(\PageIndex{149}\)

    \(\sqrt[3]{q^5}\)

    Exercise \(\PageIndex{150}\)

    \(\sqrt[5]{n^8}\)

    Answer

    \(n^{\frac{8}{5}}\)

    In the following exercises, simplify.

    Exercise \(\PageIndex{151}\)

    \(27^{−\frac{2}{3}}\)

    Exercise \(\PageIndex{152}\)

    \(64^{\frac{5}{2}}\)

    Answer

    32,768

    Exercise \(\PageIndex{153}\)

    \(36^{\frac{3}{2}}\)

    Exercise \(\PageIndex{154}\)

    \(81^{−\frac{5}{2}}\)

    Answer

    \(\frac{1}{59,049}\)

    ​​​​​​​Use the Laws of Exponents to Simplify Expressions with Rational Exponents

    In the following exercises, simplify.

    Exercise \(\PageIndex{155}\)

    \(3^{\frac{4}{5}}·3^{\frac{6}{5}}\)

    Exercise \(\PageIndex{156}\)

    \((x^6)^{\frac{4}{3}}\)

    Answer

    \(x^8\)

    Exercise \(\PageIndex{157}\)

    \(\frac{z^{\frac{5}{2}}}{z^{\frac{7}{5}}}\)

    Exercise \(\PageIndex{158}\)

    \((16s^{\frac{9}{4}})^{\frac{1}{4}}\)

    Answer

    \(2s^{\frac{9}{16}}\)

    Exercise \(\PageIndex{159}\)

    \((m^{8}n^{12})^{\frac{1}{4}}\)

    Exercise \(\PageIndex{160}\)

    \(\frac{z^{\frac{2}{3}}·z^{−\frac{1}{3}}}{z^{−\frac{5}{3}}}\)

    Answer

    \(z^2\)​​​​​​​

    Practice Test

    In the following exercises, simplify.

    Exercise \(\PageIndex{161}\)

    \(\sqrt{81+144}\)

    Exercise \(\PageIndex{162}\)

    \(\sqrt{169m^{4}n^{2}}\)

    Answer

    \(13m^{2}|n|\)

    Exercise \(\PageIndex{163}\)

    \(\sqrt{36n^{13}}\)

    Exercise \(\PageIndex{164}\)

    \(3\sqrt{13}+5\sqrt{2}+\sqrt{13}\)

    Answer

    \(4\sqrt{13}+5\sqrt{2}\)

    Exercise \(\PageIndex{165}\)

    \(5\sqrt{20}+2\sqrt{125}\)

    Exercise \(\PageIndex{166}\)

    \((3\sqrt{6y})(\sqrt{250y^3})\)

    Answer

    \(180y^2\sqrt{3}\)

    Exercise \(\PageIndex{167}\)

    \((2−5\sqrt{x})(3+\sqrt{x})\)

    Exercise \(\PageIndex{168}\)

    \((1−2\sqrt{q})^2\)

    Answer

    \(1−4\sqrt{q}+4q\)

    Exercise \(\PageIndex{169}\)

    1. \(\sqrt{a^{12}}\)
    2. \(\sqrt[3]{b^{21}}\)

    Exercise \(\PageIndex{170}\)

    1. \(\sqrt[4]{81x^{12}}\)
    2. \(\sqrt[6]{64y^{18}}\)
    Answer
    1. \(3x^3\)
    2. \(2y^3\)

    Exercise \(\PageIndex{171}\)

    \(\sqrt[6]{\frac{64r^{12}}{25r^6}}\)​​​​​​​

    Exercise \(\PageIndex{172}\)

    \(\sqrt{\frac{14y^3}{7y}}\)

    Answer

    \(y\sqrt{2}\)

    Exercise \(\PageIndex{173}\)

    \(\frac{\sqrt{256x^7}}{\sqrt{54x^2}}\)

    Exercise \(\PageIndex{174}\)

    \(\sqrt[4]{512}−2\sqrt[4]{32}\)

    Answer

    0

    Exercise \(\PageIndex{175}\)

    1. \(256^{\frac{1}{4}}\)
    2. \(243^{\frac{1}{5}}\)

    Exercise \(\PageIndex{176}\)

    \(49^{\frac{3}{2}}\)

    Answer

    343

    Exercise \(\PageIndex{177}\)

    \(25^{−\frac{5}{2}}\)

    Exercise \(\PageIndex{178}\)

    \(\frac{w^{\frac{3}{4}}}{w^{\frac{7}{4}}}\)

    Answer

    \(\frac{1}{w}\)

    Exercise \(\PageIndex{179}\)

    \((27s^{\frac{3}{5}})^{\frac{1}{3}}\)

    ​​​​​​​In the following exercises, rationalize the denominator.

    Exercise \(\PageIndex{180}\)

    \(\frac{3}{2\sqrt{6}}\)

    Answer

    \(\frac{\sqrt{6}}{4}\)

    Exercise \(\PageIndex{181}\)

    \(\frac{\sqrt{3}}{\sqrt{x}+\sqrt{5}}\)

    In the following exercises, solve.

    Exercise \(\PageIndex{182}\)

    \(3\sqrt{2x−3}−20=7\)

    Answer

    42​​​​​​​

    Exercise \(\PageIndex{183}\)

    \(\sqrt{3u−2}=\sqrt{5u+1}\)

    ​​​​​​​In the following exercise, solve.

    Exercise \(\PageIndex{184}\)

    A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.

    Answer

    6.1 seconds

    Glossary

    rational exponents
    • If \(\sqrt[n]{a}\) is a real number and \(n \ge 2\), \(a^{\frac{1}{n}}=\sqrt[n]{a}\)
    • For any positive integers m and n, \(a^{\frac{m}{n}}=(\sqrt[n]{a})^m\) and \(a^{\frac{m}{n}}=\sqrt[n]{a^m}\)
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