1.4.3: Rational Exponents
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 properties of exponents to simplify expressions with rational exponents
Before you get started, take this readiness quiz.
- Add \(\dfrac{7}{15}+\dfrac{5}{12}\).
- Simplify \((4x^{2}y^{5})^{3}\).
- Simplify \(5^{−3}\).
The \(n^{\mathrm{th}}\) root
In the previous sections we worked with the simplest radical expressions, i.e. square roots. This can be generalized as follows.
The \(n^{\mathrm{th}}\) root of a is \(b\) if \(b^n=a\). If \(n\) is even, take \(b\) to be positive and we write \(\sqrt[n]a=b\). We call \(n\) the index and \(a\) the radicand .
Evaluate:
a. \(\sqrt[4]{10000}\)
b. \(\sqrt[3]{-27}\)
c. \(\sqrt[5]{-32x^{10}y^{5}{z^{20}}}\)
Solution
a. \(10^4=10000\), so \(\sqrt[4]{10000}=10\)
b. \((-3)^3=-27\), so \(\sqrt[3]{-27}=-3\)
c. \((-2x^{2}yz^{4})^5=-32x^{10}y^{5}z^{20}\), so \(\sqrt[5]{-32x^{10}y^{5}z^{20}}=-2x^{2}yz^{4}\)
Evaluate:
a. \(\sqrt[3]{64}\)
b. \(\sqrt[5]{-243}\)
- Answer
-
a. \(4\)
b. \(-3\)
Evaluate:
a. \(\sqrt[4]{16x^8y^{12}z^4}\)
b. \(\sqrt[5]{-100000p^{-25}}\)
- Answer
-
a. \(2x^2y^{3}z\)
b. \(-\dfrac{10}{p^5}\)
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 \(\left(a^{m}\right)^{n}=a^{mn}\) when \(m\) and \(n\) are integers. Let’s assume we are now not limited to integers.
Suppose we want to find a number \(p\) such that \(\left(8^{p}\right)^{3}=8\). We will use the Power Property of Exponents to find the value of \(p\).
\(\left(8^{p}\right)^{3}=8\)
Multiply the exponents on the left.
\(8^{3p}=8\)
Write the exponent \(1\) on the right.
\(8^{3p}=8^{1}\)
Since the bases are the same, the exponents must be equal.
\(3p=1\)
Solve for \(p\).
\(p=\dfrac{1}{3}\)
So \(\left(8^{\frac{1}{3}}\right)^{3}=8\). 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 \(a^{\frac{1}{n}}=\sqrt[n]{a}\).
If \(\sqrt[n]{a}\) is a real number and \(n \geq 2\), then
\(a^{\frac{1}{n}}=\sqrt[n]{a}.\)
The denominator of the rational exponent is the index of the radical.
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.
Write as a radical expression:
a. \(x^{\frac{1}{2}}\)
b. \(y^{\frac{1}{3}}\)
c. \(z^{\frac{1}{4}}\)
- Solution
-
We want to write each expression in the form \(\sqrt[n]{a}\).
a.
\(x^{\frac{1}{2}}\) The denominator of the rational exponent is \(2\), so the index of the radical is \(2\). We do not show the index when it is \(2\). \(\sqrt{x}\) b.
\(y^{\frac{1}{3}}\) The denominator of the exponent is \(3\), so the index is \(3\). \(\sqrt[3]{y}\) c.
\(z^{\frac{1}{4}}\) The denominator of the exponent is \(4\), so the index is \(4\).
\(\sqrt[4]{z}\)
Write as a radical expression:
a. \(t^{\frac{1}{2}}\)
b. \(m^{\frac{1}{3}}\)
c. \(r^{\frac{1}{4}}\)
- Answer
-
a. \(\sqrt{t}\)
b. \(\sqrt[3]{m}\)
c. \(\sqrt[4]{r}\)
Write as a radical expression:
a. \(b^{\frac{1}{6}}\)
b. \(z^{\frac{1}{5}}\)
c. \(p^{\frac{1}{4}}\)
- Answer
-
a. \(\sqrt[6]{b}\)
b. \(\sqrt[5]{z}\)
c. \(\sqrt[4]{p}\)
In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.
Write with a rational exponent:
\[\sqrt{5y}\]
- Solution
-
We want to write each radical in the form \(a^{\frac{1}{n}}\)
\(\sqrt{5y}\)
No index is shown, so it is \(2\).
The denominator of the exponent will be \(2\).
Put parentheses around the entire expression \(5y\).
\((5 y)^{\frac{1}{2}}\)
Write with a rational exponent:
\[\sqrt{10m}\]
- Answer
-
\((10 m)^{\frac{1}{2}}\)
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
Simplify:
a. \(25^{\frac{1}{2}}\)
b. \(64^{\frac{1}{3}}\)
c. \(256^{\frac{1}{4}}\)
- Solution
-
a.
\(25^{\frac{1}{2}}\) Rewrite as a square root. \(\sqrt{25}\) Simplify. \(5\) b.
\(64^{\frac{1}{3}}\) Rewrite as a cube root. \(\sqrt[3]{64}\) Recognize \(64\) is a perfect cube. \(\sqrt[3]{4^{3}}\) Simplify. \(4\) c.
\(256^{\frac{1}{4}}\) Rewrite as a fourth root. \(\sqrt[4]{256}\) Recognize \(256\) is a perfect fourth power. \(\sqrt[4]{4^{4}}\) Simplify. \(4\)
Simplify:
a. \(36^{\frac{1}{2}}\)
b. \(8^{\frac{1}{3}}\)
c. \(16^{\frac{1}{4}}\)
- Answer
-
a. \(6\)
b. \(2\)
c. \(2\)
Simplify:
a. \(100^{\frac{1}{2}}\)
b. \(27^{\frac{1}{3}}\)
c. \(81^{\frac{1}{4}}\)
- Answer
-
a. \(10\)
b. \(3\)
c. \(3\)
Be careful of the placement of the negative signs in the next example. We will need to use the definition \(a^{-n}=\dfrac{1}{a^{n}}\) in one case.
Simplify:
a. \((-16)^{\frac{1}{4}}\)
b. \(-16^{\frac{1}{4}}\)
c. \((16)^{-\frac{1}{4}}\)
- Solution
-
a.
\((-16)^{\frac{1}{4}}\) Rewrite as a fourth root. \(\sqrt[4]{-16}\) Simplify. \(\sqrt[4]{(-2)^{4}}\) No real solution
b.
\(-16^{\frac{1}{4}}\) The exponent only applies to the \(16\). Rewrite as a fourth root. \(-\sqrt[4]{16}\) Rewrite \(16\) as \(2^{4}\). \(-\sqrt[4]{2^{4}}\) Simplify. \(-2\) c.
\((16)^{-\frac{1}{4}}\) Rewrite using the definition \(a^{-n}=\dfrac{1}{a^{n}}\). \(\dfrac{1}{(16)^{\frac{1}{4}}}\) Rewrite as a fourth root. \(\dfrac{1}{\sqrt[4]{16}}\) Rewrite \(16\) as \(2^{4}\). \(\dfrac{1}{\sqrt[4]{2^{4}}}\) Simplify. \(\dfrac{1}{2}\)
Simplify:
a. \((-64)^{-\frac{1}{2}}\)
b. \(-64^{\frac{1}{2}}\)
c. \((64)^{-\frac{1}{2}}\)
- Answer
-
a. No real solution
b. \(-8\)
c. \(\dfrac{1}{8}\)
Simplify:
a. \((-256)^{\frac{1}{4}}\)
b. \(-256^{\frac{1}{4}}\)
c. \((256)^{-\frac{1}{4}}\)
- Answer
-
a. No real solution
b. \(-4\)
c. \(\dfrac{1}{4}\)
Simplify Expressions with \(a^{\frac{m}{n}}\)
We can look at \(a^{\frac{m}{n}}\) in two ways. Remember the Power Property tells us to multiply the exponents and so \(\left(a^{\frac{1}{n}}\right)^{m}\) and \(\left(a^{m}\right)^{\frac{1}{n}}\) both equal \(a^{\frac{m}{n}}\). If we write these expressions in radical form, we get
\(a^{\frac{m}{n}}=\left(a^{\frac{1}{n}}\right)^{m}=(\sqrt[n]{a})^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\left(a^{m}\right)^{^{\frac{1}{n}}}=\sqrt[n]{a^{m}}\)
This leads us to the following definition.
For any positive integers \(m\) and \(n\),
\(a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^{m} \quad \text { and } \quad a^{\frac{m}{n}}=\sqrt[n]{a^{m}}.\)
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, before raising it to the power indicated.
Write with a rational exponent:
a. \(\sqrt{y^{3}}\)
b. \((\sqrt[3]{2 x})^{4}\)
c. \(\sqrt{\left(\dfrac{3 a}{4 b}\right)^{3}}\)
- Solution
-
We want to use \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) to write each radical in the form \(a^{\frac{m}{n}}\)
a.
\(\sqrt{y^{3}}\) When the index is missing, it is implicitly \(2\). \(\sqrt[2]{y^{3}}\) \(m=3, n=2\) in \(\sqrt[n]{a^{m}}=a^{\frac{m}{n}}\) \(y^{\frac{3}{2}}\) b.
c.
Write with a rational exponent:
a. \(\sqrt{x^{5}}\)
b. \((\sqrt[4]{3 y})^{3}\)
c. \(\sqrt{\left(\dfrac{2 m}{3 n}\right)^{5}}\)
- Answer
-
a. \(x^{\frac{5}{2}}\)
b. \((3 y)^{\frac{3}{4}}\)
c. \(\left(\dfrac{2 m}{3 n}\right)^{\frac{5}{2}}\)
Write with a rational exponent:
a. \(\sqrt[5]{a^{2}}\)
b. \(\left(\sqrt[3]{5 a b}\right)^{5}\)
c. \(\sqrt{\left(\dfrac{7 x y}{z}\right)^{3}}\)
- Answer
-
a. \(a^{\frac{2}{5}}\)
b. \((5 a b)^{\frac{5}{3}}\)
c. \(\left(\dfrac{7 x y}{z}\right)^{\frac{3}{2}}\)
Remember that \(a^{-n}=\dfrac{1}{a^{n}}\). The negative sign in the exponent does not change the sign of the expression.
Simplify:
a. \(125^{\frac{2}{3}}\)
b. \(16^{-\frac{3}{2}}\)
c. \(32^{-\frac{2}{5}}\)
- Solution
-
We will rewrite the expression as a radical first using the defintion, \(a^{\frac{m}{n}}=(\sqrt[n]{a})^{m}\). 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.
a.
\(125^{\frac{2}{3}}\) The power of the radical is the numerator of the exponent, \(2\). The index of the radical is the denominator of the exponent, \(3\). \(\left(\sqrt[3]{125}\right)^{2}\) Simplify. \((5)^{2}\) \(25\) b.
\(16^{-\frac{3}{2}}\) We will rewrite each expression first using \(a^{-n}=\dfrac{1}{a^{n}}\) and then change to radical form. \(\dfrac{1}{16^{\frac{3}{2}}}\) Change to radical form. The power of the radical is the numerator of the exponent, \(3\). The index is the denominator of the exponent, \(2\). \(\dfrac{1}{\left(\sqrt{16}\right)^{3}}\) Simplify. \(\dfrac{1}{4^{3}}\)
\(\dfrac{1}{64}\)
c.
\(32^{-\frac{2}{5}}\) Rewrite using \(a^{-n}=\dfrac{1}{a^{n}}\). \(\dfrac{1}{32^{\frac{2}{5}}}\) Change to radical form. \(\dfrac{1}{(\sqrt[5]{32})^{2}}\) Rewrite the radicand as a power. \(\dfrac{1}{\left(\sqrt[5]{2^{5}}\right)^{2}}\) Simplify. \(\dfrac{1}{2^{2}}\)
\(\dfrac{1}{4}\)
Simplify:
a. \(27^{\frac{2}{3}}\)
b. \(81^{-\frac{3}{2}}\)
c. \(16^{-\frac{3}{4}}\)
- Answer
-
a. \(9\)
b. \(\dfrac{1}{729}\)
c. \(\dfrac{1}{8}\)
Simplify:
a. \(4^{\frac{3}{2}}\)
b. \(27^{-\frac{2}{3}}\)
c. \(625^{-\frac{3}{4}}\)
- Answer
-
a. \(8\)
b. \(\dfrac{1}{9}\)
c. \(\dfrac{1}{125}\)
Simplify:
a. \(-25^{\frac{3}{2}}\)
b. \(-25^{-\frac{3}{2}}\)
c. \((-25)^{\frac{3}{2}}\)
- Solution
-
a.
\(-25^{\frac{3}{2}}\) Rewrite in radical form. \(-(\sqrt{25})^{3}\) Simplify the radical. \(-(5)^{3}\) Simplify. \(-125\) b.
\(-25^{-\frac{3}{2}}\) Rewrite using \(a^{-n}=\dfrac{1}{a^{n}}\). \(-\left(\dfrac{1}{25^{\frac{3}{2}}}\right)\) Rewrite in radical form. \(-\left(\dfrac{1}{\left(\sqrt{25}\right)^{3}}\right)\) Simplify the radical. \(-\left(\dfrac{1}{5^{3}}\right)\) Simplify. \(-\dfrac{1}{125}\) c.
\((-25)^{\frac{3}{2}}\) Rewrite in radical form. \(\left(\sqrt{-25}\right)^{3}\)
There is no real number whose square root is \(-25\).
Not a real number.
Simplify:
a. \(-16^{\frac{3}{2}}\)
b. \(-16^{-\frac{3}{2}}\)
c. \((-16)^{-\frac{3}{2}}\)
- Answer
-
a. \(-64\)
b. \(-\dfrac{1}{64}\)
c. Not a real number
Simplify:
a. \(-81^{\frac{3}{2}}\)
b. \(-81^{-\frac{3}{2}}\)
c. \((-81)^{-\frac{3}{2}}\)
- Answer
-
a. \(-729\)
b. \(-\dfrac{1}{729}\)
c. Not a real number
Use the Properties of Exponents to Simplify Expressions with Rational Exponents
The same properties of exponents that we have already used for integers also apply to rational exponents. We will not show this here, but doing some examples will convince you. For example,
\[16^{\frac{1}{2}}\cdot 16^{\frac{3}{4}} = 4\cdot 8 =32 \nonumber\]
and
\[16^{\frac{1}{2}+\frac{3}{4}} = 16^{\frac{5}{4}} =32. \nonumber\]
So \(16^{\frac{1}{2}}\cdot 16^{\frac{3}{4}} = 16^{\frac{1}{2}+\frac{3}{4}}\).
We will list the Properties of Exponents below to have them for reference as we simplify expressions.
If \(a\) and \(b\) are real numbers and \(m\) and \(n\) are rational numbers, then
Product Property
\(a^{m} \cdot a^{n}=a^{m+n}\)
Power Property
\(\left(a^{m}\right)^{n}=a^{m n}\)
Product to a Power
\((a b)^{m}=a^{m} b^{m}\)
Quotient Property
\(\dfrac{a^{m}}{a^{n}}=a^{m-n}, a \neq 0\)
Zero Exponent Definition
\(a^{0}=1, a \neq 0\)
Quotient to a Power Property
\(\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}, b \neq 0\)
Negative Exponent Property
\(a^{-n}=\dfrac{1}{a^{n}}, a \neq 0\)
We will apply these properties in the next example.
Simplify:
a. \(x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}\)
b. \(\left(z^{9}\right)^{\frac{2}{3}}\)
c. \(\dfrac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\)
- Solution
-
a.
The Product Property tells us that when we multiple the same base, we add the exponents. \(x^{\frac{1}{2}} \cdot x^{\frac{5}{6}}\) The bases are the same, so we add the exponents. \(x^{\frac{1}{2}+\frac{5}{6}}\) Add the fractions. \(x^{\frac{8}{6}}\) Simplify the exponent. \(x^{\frac{4}{3}}\) b.
The Power Property tells us that when we raise a power to a power, we multiple the exponents. \(\left(z^{9}\right)^{\frac{2}{3}}\) To raise a power to a power, we multiple the exponents. \(z^{9 \cdot \frac{2}{3}}\) Simplify. \(z^{6}\) c.
The Quotient Property tells us that when we divide with the same base, we subtract the exponents. \(\frac{x^{\frac{1}{3}}}{x^{\frac{5}{3}}}\) To divide with the same base, we subtract the exponents. \(\dfrac{1}{x^{\frac{5}{3}-\frac{1}{3}}}\) Simplify. \(\dfrac{1}{x^{\frac{4}{3}}}\)
Simplify:
a. \(x^{\frac{1}{6}} \cdot x^{\frac{4}{3}}\)
b. \(\left(x^{6}\right)^{\frac{4}{3}}\)
c. \(\dfrac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}\)
- Answer
-
a. \(x^{\frac{3}{2}}\)
b. \(x^{8}\)
c. \(\dfrac{1}{x}\)
Simplify:
a. \(y^{\frac{3}{4}} \cdot y^{\frac{5}{8}}\)
b. \(\left(m^{9}\right)^{\frac{2}{9}}\)
c. \(\dfrac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}\)
- Answer
-
a. \(y^{\frac{11}{8}}\)
b. \(m^{2}\)
c. \(\dfrac{1}{d}\)
- How is the square root related to rational exponents?
- What about a cube root?
- Show two different algebraic methods to simplify \(4^{\frac{3}{2}}\). Explain all your steps.
- Explain why the expression \((-16)^{\frac{3}{2}}\) cannot be evaluated.
- Explain why \(x^{\frac12}=\sqrt{x}\).
- Give and example of the rules \((ab)^n=a^nb^n\) and \(a^na^m=a^{n+m}\) with rational exponents.
Simplify \((-27b^9)^{1/3}\).
Key Concepts
-
Rational Exponent \(a^{\frac{1}{n}}\)
- If \(\sqrt[n]{a}\) is a real number and \(n≥2\), then \(a^{\frac{1}{n}}=\sqrt[n]{a}\).
-
Rational Exponent \(a^{\frac{m}{n}}\)
-
For any positive integers \(m\) and \(n\),
\(a^{\frac{m}{n}}=\left(\sqrt[n]{a}\right)^{m} \text { and } a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\)
-
For any positive integers \(m\) and \(n\),
-
Properties of Exponents
-
If \(a, b\) are real numbers and \(m, n\) are rational numbers, then
- Product Property \(\qquad a^{m} \cdot a^{n}=a^{m+n}\)
- Power Property \(\qquad\left(a^{m}\right)^{n}=a^{mn}\)
- Product to a Power \(\qquad(a b)^{m}=a^{m} b^{m}\)
- Quotient Property \(\qquad\dfrac{a^{m}}{a^{n}}=a^{m-n}, a \neq 0\)
- Zero Exponent Definition \(\qquad a^{0}=1, a \neq 0\)
- Quotient to a Power Property \(\qquad\left(\dfrac{a}{b}\right)^{m}=\dfrac{a^{m}}{b^{m}}, b \neq 0\)
- Negative Exponent Property \(\qquad a^{-n}=\dfrac{1}{a^{n}}, a \neq 0\)
-
If \(a, b\) are real numbers and \(m, n\) are rational numbers, then