9.3: Simplifying Square Root Expressions
To begin our study of the process of simplifying a square root expression, we must note three facts: one fact concerning perfect squares and two concerning properties of square roots.
Perfect Squares
Rea numbers that are squares of rational numbers are called perfect squares . The numbers \(25\) and \(\dfrac{1}{4}\) are examples of perfect squares since \(25 = 5^2\) and \(\dfrac{1}{4} = (\dfrac{1}{2})^2\), and \(5\) and \(\dfrac{1}{2}\) are rational numbers. The number \(2\) is not a perfect square since \(2 = (\sqrt{2})^2\) and \(\sqrt{2}\) is not a rational number.
Although we will not make a detailed study of irrational numbers, we will make the following observation:
Any indicated square root whose radicand is not a perfect square number is an irrational number.
The numbers \(\sqrt{6}, \sqrt{15}\) and \(\sqrt{\dfrac{3}{4}}\) are each irrational since each radicand \(6, 15, \dfrac{3}{4}\) is not a perfect square.
The Product Property of Square Roots
Notice that
\(\begin{array}{flushleft}
\sqrt{9 \cdot 4} &= \sqrt{36} &= 6 & \text{ and }\\
\sqrt{9} \sqrt{4} &= 3 \cdot 2 &= 6
\end{array}\)
This suggests that in general, if \(x\) and \(y\) are positive real numbers,
\(\sqrt{xy} = \sqrt{x} \sqrt{y}\)
The square root of the product is the product of the square roots.
The Quotient Property of Square Roots
We can suggest a similar rule for quotients. Notice that
\(\sqrt{\dfrac{36}{4}} = \sqrt{9} = 3\) and
\(\dfrac{\sqrt{36}}{\sqrt{4}} = \dfrac{6}{2} = 3\).
Since both \(\dfrac{36}{4}\) and \(\dfrac{\sqrt{36}}{\sqrt{4}}\) equal \(3\), it must be that
\(\sqrt{\dfrac{36}{4}} = \dfrac{\sqrt{36}}{\sqrt{4}}\)
This suggests that in general, if \(x\) and \(y\) are positive real numbers,
\(\sqrt{\dfrac{x}{y}} = \dfrac{\sqrt{x}}{\sqrt{y}}, y \not = 0\).
The square root of the quotient is the quotient of the square roots.
It is extremely important to remeber that
\(\sqrt{x + y} \not = \sqrt{x} + \sqrt{y}\) or \(\sqrt{x - y} \not = \sqrt{x} - \sqrt{y}\)
For example, notice that \(\sqrt{16 + 9} = \sqrt{25} = 5\), but \(\sqrt{16} + \sqrt{9} = 4 + 3 = 7\)
We shall study the process of simplifying a square root expresion by distinguishing between two types of square roots: square roots not involving a fraction and square roots involving a fraction.
Square Roots Not Involving Fractions
A square root that does not involve fractions is in the simplified form if there is no perfect square in the radicand.
The square roots \(\sqrt{x},. \sqrt{ab}, \sqrt{5mn}, \sqrt{2(a+5)}\) are in simplified form since none of the radicands contains a perfect square.
The square roots \(\sqrt{x^2}, \sqrt{a^3}=\sqrt{a^2a}\) are not in simplified form since each radicand contains a perfect square.
To simplify a square root expression that does not involve a fraction, we can use the following two rules:
- If a factor of the radicand contains a variable with an even exponent, the square root is obtained by dividing the exponent by 2.
- If a factor of the radicand contains a variable with an odd exponent, the square root is obtained by first factoring the variable factor into two factors so that one has an even exponent and the other has an exponent of 1, then using the product property of square roots.
Sample Set A
Simplify each square root.
\(\sqrt{a^4}\). The exponent is even: \(\dfrac{4}{2} = 2\). The exponent on the square root is \(2\).
\(\sqrt{a^4} = a^2\)
\(\sqrt{a^6b^{10}}\). Both exponents are even: \(\dfrac{6}{2} = 3\) and \(\dfrac{10}{2} = 5\). The exponent on the square root of \(a^6\) is \(3\). The exponent on the square root if \(b^{10}\) is \(5\).
\(\sqrt{a^6gb^{10}} = a^3b^5\)
\(\sqrt{y^5}\). The exponent is odd: \(y^5 = y^4y\). The
\(\sqrt{y^5} = \sqrt{y^4y} = \sqrt{y^4} \sqrt{y} = y^2 \sqrt{y}\)
\(\begin{array}{flushleft}
\sqrt{36a^7b^{11}c^{20}} &= \sqrt{6^2a^6ab^{10}bc^{20}} & a^7 = a^6a, b^{11} = b^{10}b\\
&= \sqrt{6^2a^6b^{10}c^{20} \cdot ab} & \text{ by the commutative property of multiplication }\\
&= \sqrt{6^2a^6b^{10}c^{20}} \sqrt{ab} & \text{ by the product property of square roots }\\
&= 6a^3b^5c^{10} \sqrt{ab}
\end{array}\)
\(\begin{array}{flushleft}
\sqrt{49x^8y^3(a-1)^6} &= \sqrt{7^2x^8y^2y(a-1)^6}\\
&= \sqrt{7^2x^8y^2(a-1)^6} \sqrt{y}\\
&= 7x^4y(a-1)^3 \sqrt{y}
\end{array}\)
\(\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{5^2 \cdot 3}= \sqrt{5^2} \sqrt{3} = 5 \sqrt{3}\)
Practice Set A
Simplify each square root.
\(\sqrt{m^8}\)
- Answer
-
\(m^4\)
\(\sqrt{h^{14}k^{22}}\)
- Answer
-
\(h^7k^{11}\)
\(\sqrt{81a^{12}b^6c^{38}}\)
- Answer
-
\(9a^6b^3c^{19}\)
\(\sqrt{144x^4y^{80}(b+5)^{16}}\)
- Answer
-
\(12x^2y^{40}(b+5)^8\)
\(\sqrt{w^5}\)
- Answer
-
\(w^2 \sqrt{w}\)
\(\sqrt{w^7z^3k^{13}}\)
- Answer
-
\(w^3zk^6 \sqrt{wzk}\)
\(\sqrt{27a^3b^4c^5d^6}\)
- Answer
-
\(3ab^2c^2d^3 \sqrt{3ac}\)
\(\sqrt{180m^4n^{15}9a-12)^{15}}\)
- Answer
-
\(6m^2n^7(a-12)^7 \sqrt{5n(a-12)}\)
Square Roots Involving Fractions
A square root expression is in simplified form if there are
- no perfect squares in the radicand,
- no fractions in the radicand, or
- no square root expressions in the denominator.
The square root expressions \(\sqrt{5a}, \dfrac{4\sqrt{3xy}}{5}\), and \(\dfrac{11m^2n \sqrt{a-4}}{2x^2}\) are in simplified form
The square root expressions \(\sqrt{\dfrac{3x}{8}}, \sqrt{\dfrac{4a^4b^3}{5}}\), and \(\dfrac{2y}{\sqrt{3x}}\) are not in simplified form.
To simplify the square root expression \(\sqrt{\dfrac{x}{y}}\),
- Write the expression as \(\dfrac{\sqrt{x}}{\sqrt{y}}\) using the rule \(\sqrt{\dfrac{x}{y}} = \dfrac{\sqrt{x}}{\sqrt{y}}\).
- Multiply the fraction by 1 in the form of \(\dfrac{\sqrt{y}}{\sqrt{y}}\).
- Simplify the remaining fraction, \(\dfrac{\sqrt{xy}}{y}\).
Rationalizing the Denominator
The process involved in step 2 is called rationalizing the denominator. This process removes square root expressions from the denominator using the fact that \((\sqrt{y})(\sqrt{y}) = y\).
Sample Set B
Simplify each square root.
\(\sqrt{\dfrac{9}{25}} = \dfrac{\sqrt{9}}{\sqrt{25}} = \dfrac{3}{5}\)
\(\sqrt{\dfrac{3}{5}}=\dfrac{\sqrt{3}}{\sqrt{5}}=\dfrac{\sqrt{3}}{\sqrt{5}} \cdot \dfrac{\sqrt{5}}{\sqrt{5}}=\dfrac{\sqrt{15}}{5}\)
\(\sqrt{\dfrac{9}{8}}=\dfrac{\sqrt{9}}{\sqrt{8}}=\dfrac{\sqrt{9}}{\sqrt{8}} \cdot \dfrac{\sqrt{8}}{\sqrt{8}}=\dfrac{3 \sqrt{8}}{8}=\dfrac{3 \sqrt{4 \cdot 2}}{8}=\dfrac{3 \sqrt{4} \sqrt{2}}{8}=\dfrac{3 \cdot 2 \sqrt{2}}{8}=\dfrac{3 \sqrt{2}}{4}\)
\(\sqrt{\dfrac{k^{2}}{m^{3}}}=\dfrac{\sqrt{k^{2}}}{\sqrt{m^{3}}}=\dfrac{k}{\sqrt{m^{3}}}=\dfrac{k}{\sqrt{m^{2} m}}=\dfrac{k}{\sqrt{m^{2} \sqrt{m}}}=\dfrac{k}{m \sqrt{m}}=\dfrac{k}{m \sqrt{m}} \cdot \dfrac{\sqrt{m}}{\sqrt{m}}=\dfrac{k \sqrt{m}}{m \sqrt{m} \sqrt{m}}=\dfrac{k \sqrt{m}}{m \cdot m}=\dfrac{k \sqrt{m}}{m^{2}}\)
\(\begin{array}{flushleft}
\sqrt{x^2 - 8x + 16} &= \sqrt{(x-4)^2}\\
&= x-4
\end{array}\)
Practice Set B
Simplify each square root.
\(\sqrt{\dfrac{81}{25}}\)
- Answer
-
\(\dfrac{9}{5}\)
\(\sqrt{\dfrac{2}{7}}\)
- Answer
-
\(\dfrac{\sqrt{14}}{7}\)
\(\sqrt{\dfrac{4}{5}}\)
- Answer
-
\(\dfrac{2 \sqrt{5}}{5}\)
\(\sqrt{\dfrac{10}{4}}\)
- Answer
-
\(\dfrac{\sqrt{10}}{2}\)
\(\sqrt{\dfrac{9}{4}}\)
- Answer
-
\(\dfrac{3}{2}\)
\(\sqrt{\dfrac{a^3}{6}}\)
- Answer
-
\(\dfrac{a \sqrt{6a}}{6}\)
\(\sqrt{\dfrac{y^4}{x^3}}\)
- Answer
-
\(\dfrac{y^2 \sqrt{x}}{x^2}\)
\(\sqrt{\dfrac{32a^5}{b^7}}\)
- Answer
-
\(\dfrac{4a^2 \sqrt{2ab}}{b^4}\)
\(\sqrt{(x+9)^2}\)
- Answer
-
\(x+9\)
\(\sqrt{x^2 + 14x + 49}\)
- Answer
-
\(x+7\)
Exercises
For the following problems, simplify each of the radical expressions.
\(\sqrt{8b^2}\)
- Answer
-
\(2b \sqrt{2}\)
\(\sqrt{20a^2}\)
\(\sqrt{24x^4}\)
- Answer
-
\(2x^2 \sqrt{6}\)
\(\sqrt{27y^6}\)
\(\sqrt{a^5}\)
- Answer
-
\(a^2\sqrt{a}\)
\(\sqrt{m^7}\)
\(\sqrt{x^{11}}\)
- Answer
-
\(x^5 \sqrt{x}\)
\(\sqrt{y^{17}}\)
\(\sqrt{36n^9}\)
- Answer
-
\(6n^4 \sqrt{n}\)
\(\sqrt{49x^{13}}\)
\(\sqrt{100x^5y^{11}}\)
- Answer
-
\(10x^2y^5 \sqrt{xy}\)
\(\sqrt{64a^7b^3}\)
\(5 \sqrt{16m^6n^7}\)
- Answer
-
\(20m^3n^3 \sqrt{n}\)
\(8 \sqrt{9a^4b^{11}}\)
\(3 \sqrt{16x^3}\)
- Answer
-
\(12x \sqrt{x}\)
\(8 \sqrt{25y^3}\)
\(\sqrt{12a^4}\)
- Answer
-
\(2a^2 \sqrt{3}\)
\(\sqrt{32x^7}\)
- Answer
-
\(4x^3 \sqrt{2x}\)
\(\sqrt{12y^{13}}\)
\(\sqrt{50a^3b^9}\)
- Answer
-
\(5ab^4 \sqrt{2ab}\)
\(\sqrt{48p^{11}q^5}\)
\(4 \sqrt{18a^5b^{17}}\)
- Answer
-
\(12a^2b^8 \sqrt{2ab}\)
\(8 \sqrt{108x^{21}y^3}\)
\(-4 \sqrt{75a^4b^6}\)
- Answer
-
\(-20a^2b^3 \sqrt{3}\)
\(-6 \sqrt{72x^2y^4z^{10}}\)
\(-\sqrt{b^{12}}\)
- Answer
-
\(-b^6\)
\(- \sqrt{c^{18}}\)
\(\sqrt{a^2b^2c^2}\)
- Answer
-
\(abc\)
\(\sqrt{4x^2y^2z^2}\)
\(- \sqrt{9a^2b^3}\)
- Answer
-
\(-3ab \sqrt{b}\)
\(- \sqrt{16x^4y^5}\)
\(\sqrt{m^6n^8p^{12}q^{20}}\)
- Answer
-
\(m^3n^4p^6q^{10}\)
\(\sqrt{r^2}\)
\(\sqrt{p^2}\)
- Answer
-
\(p\)
\(\sqrt{\dfrac{1}{4}}\)
\(\sqrt{\dfrac{1}{16}}\)
- Answer
-
\(\dfrac{1}{4}\)
\(\sqrt{\dfrac{4}{25}}\)
\(\sqrt{\dfrac{9}{49}}\)
- Answer
-
\(\dfrac{3}{7}\)
\(\dfrac{5 \sqrt{8}}{\sqrt{3}}\)
\(\dfrac{2 \sqrt{32}}{\sqrt{3}}\)
- Answer
-
\(\dfrac{8 \sqrt{6}}{3}\)
\(\sqrt{\dfrac{5}{6}}\)
\(\sqrt{\dfrac{2}{7}}\)
- Answer
-
\(\dfrac{\sqrt{14}}{7}\)
\(\sqrt{\dfrac{3}{10}}\)
\(\sqrt{\dfrac{4}{3}}\)
- Answer
-
\(\dfrac{2 \sqrt{3}}{3}\)
\(-\sqrt{\dfrac{2}{5}}\)
\(-\sqrt{\dfrac{3}{10}}\)
- Answer
-
\(-\dfrac{\sqrt{30}}{10}\)
\(\sqrt{\dfrac{16a^2}{5}}\)
\(\sqrt{\dfrac{24a^5}{7}}\)
- Answer
-
\(\dfrac{2a^2 \sqrt{42a}}{7}\)
\(\sqrt{\dfrac{72x^2y^3}{5}}\)
\(\sqrt{\dfrac{2}{a}}\)
- Answer
-
\(\dfrac{\sqrt{2a}}{a}\)
\(\sqrt{\dfrac{5}{b}}\)
\(\sqrt{\dfrac{6}{x^3}}\)
- Answer
-
\(\dfrac{\sqrt{6x}}{x^2}\)
\(\sqrt{\dfrac{12}{y^5}}\)
\(\sqrt{\dfrac{49x^2y^5z^9}{25a^3b^{11}}}\)
- Answer
-
\(\dfrac{7 x y^{2} z^{4} \sqrt{a b y z}}{5 a^{2} b^{6}}\)
\(\sqrt{\dfrac{27 x^{6} y^{15}}{3^{3} x^{3} y^{5}}}\)
\(\sqrt{(b+2)^4}\)
- Answer
-
\((b+2)^2\)
\(\sqrt{(a-7)^8}\)
\(\sqrt{(x+2)^6}\)
- Answer
-
\((x+2)^3\)
\(\sqrt{(x+2)^2(x+1)^2}\)
\(\sqrt{(a-3)^4(a-1)^2}\)
- Answer
-
\((a-3)^2(a-1)\)
\(\sqrt{(b+7)^8(b-7)^6}\)
\(\sqrt{a^2 - 10a + 25}\)
- Answer
-
\((a-5)\)
\(\sqrt{b^2 + 6b + 9}\)
\(\sqrt{(a^2 - 2a + 1)^4}\)
- Answer
-
\((a-1)^4\)
\(\sqrt{(x^2 + 2x + 1)^{12}}\)
Exercises For Review
Solve the inequality \(3(a + 2) \le 2(3a + 4)\)
- Answer
-
\(a \ge -\dfrac{2}{3}\)
Graph the inequality \(6x \le 5(x+1) - 6\)
Supply the missing words. When looking at a graph from left-to-right, lines with _______ slope rise, while lines with __________ slope fall.
- Answer
-
positive; negative
Simplify the complex fraction: \(\dfrac{5+\frac{1}{x}}{5-\frac{1}{x}}\)
Simplify \(\sqrt{121x^4w^6z^8}\) by removing the radical sign.
- Answer
-
\(11x^2w^3z^4\)