Simplifying square root expressions

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To simplify a square root expression that does not involve a fraction, we can use the following two rules:

Simplifying square roots without fractions

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: $\frac{4}{2} = 2.$ The exponent on the square root is 2.

$\sqrt{a^{4}} = a^{2}$

$\sqrt{a^{6} b^{10}} .$ Both exponents are even: $\frac{6}{2} = 3$ and $\frac{10}{2} = 5.$ The exponent on the square root of $a^{6}$ is 3. The exponent on the square root if $b^{10}$ is 5.

$\sqrt{a^{6} b^{10}} = a^{3} b^{5}$

$\sqrt{y^{5}} .$ The exponent is odd: $y^{5} = y^{4} y .$ Then

$\sqrt{y^{5}} = \sqrt{y^{4} y} = \sqrt{y^{4}} \sqrt{y} = y^{2} \sqrt{y}$

$\begin{matrix} \sqrt{36 a^{7} b^{11} c^{20}} & = & \sqrt{6^{2} a^{6} a b^{10} b c^{20}} & a^{7} = a^{6} a, b^{11} = b^{10} b \\ = & \sqrt{6^{2} a^{6} b^{10} c^{20} \cdot a b} & \begin{array}{l} by the commutative property of multiplication. \end{array} \\ = & \sqrt{6^{2} a^{6} b^{10} c^{20}} \sqrt{a b} & \begin{array}{l} by the product property of square roots. \end{array} \\ = & 6 a^{3} b^{5} c^{10} \sqrt{a b} \end{matrix}$

$\begin{matrix} \sqrt{49 x^{8} y^{3} {(a - 1)}^{6}} & = & \sqrt{7^{2} x^{8} y^{2} y {(a - 1)}^{6}} \\ = & \sqrt{7^{2} x^{8} y^{2} {(a - 1)}^{6}} \sqrt{y} \\ = & 7 x^{4} y {(a - 1)}^{3} \sqrt{y} \end{matrix}$

$\sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{5^{2} \cdot 3} = \sqrt{5^{2}} \sqrt{3} = 5 \sqrt{3}$

Practice set a

Simplify each square root.

$\sqrt{m^{8}}$

$m^{4}$

$\sqrt{h^{14} k^{22}}$

$h^{7} k^{11}$

$\sqrt{81 a^{12} b^{6} c^{38}}$

$9 a^{6} b^{3} c^{19}$

$\sqrt{144 x^{4} y^{80} {(b + 5)}^{16}}$

$12 x^{2} y^{40} {(b + 5)}^{8}$

$\sqrt{w^{5}}$

$w^{2} \sqrt{w}$

$\sqrt{w^{7} z^{3} k^{13}}$

$w^{3} z k^{6} \sqrt{w z k}$

$\sqrt{27 a^{3} b^{4} c^{5} d^{6}}$

$3 a b^{2} c^{2} d^{3} \sqrt{3 a c}$

$\sqrt{180 m^{4} n^{15} {(a - 12)}^{15}}$

$6 m^{2} n^{7} {(a - 12)}^{7} \sqrt{5 n (a - 12)}$

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
3. no square root expressions in the denominator.

The square root expressions $\sqrt{5 a}, \frac{4 \sqrt{3 x y}}{5},$ and $\frac{11 m^{2} n \sqrt{a - 4}}{2 x^{2}}$ are in simplified form.

The square root expressions $\sqrt{\frac{3 x}{8}}, \sqrt{\frac{4 a^{4} b^{3}}{5}},$ and $\frac{2 y}{\sqrt{3 x}}$ are not in simplified form.

Simplifying square roots with fractions

To simplify the square root expression

\sqrt{\frac{x}{y}},

Write the expression as $\frac{\sqrt{x}}{\sqrt{y}}$ using the rule $\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}} .$
Multiply the fraction by 1 in the form of $\frac{\sqrt{y}}{\sqrt{y}} .$
Simplify the remaining fraction, $\frac{\sqrt{x y}}{y} .$

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.

$\sqrt{\frac{9}{25}} = \frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5}$

$\sqrt{\frac{3}{5}} = \frac{\sqrt{3}}{\sqrt{5}} = \frac{\sqrt{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{15}}{5}$

$\sqrt{\frac{9}{8}} = \frac{\sqrt{9}}{\sqrt{8}} = \frac{\sqrt{9}}{\sqrt{8}} \cdot \frac{\sqrt{8}}{\sqrt{8}} = \frac{3 \sqrt{8}}{8} = \frac{3 \sqrt{4 \cdot 2}}{8} = \frac{3 \sqrt{4} \sqrt{2}}{8} = \frac{3 \cdot 2 \sqrt{2}}{8} = \frac{3 \sqrt{2}}{4}$

$\sqrt{\frac{k^{2}}{m^{3}}} = \frac{\sqrt{k^{2}}}{\sqrt{m^{3}}} = \frac{k}{\sqrt{m^{3}}} = \frac{k}{\sqrt{m^{2} m}} = \frac{k}{\sqrt{m^{2}} \sqrt{m}} = \frac{k}{m \sqrt{m}} = \frac{k}{m \sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}} = \frac{k \sqrt{m}}{m \sqrt{m} \sqrt{m}} = \frac{k \sqrt{m}}{m \cdot m} = \frac{k \sqrt{m}}{m^{2}}$

$\begin{matrix} \sqrt{x^{2} - 8 x + 16} & = & \sqrt{{(x - 4)}^{2}} \\ = & x - 4 \end{matrix}$

Practice set b

Simplify each square root.

$\sqrt{\frac{81}{25}}$

$\frac{9}{5}$

$\sqrt{\frac{2}{7}}$

$\frac{\sqrt{14}}{7}$

$\sqrt{\frac{4}{5}}$

$\frac{2 \sqrt{5}}{5}$

$\sqrt{\frac{10}{4}}$

$\frac{\sqrt{10}}{2}$

$\sqrt{\frac{9}{4}}$

$\frac{3}{2}$

$\sqrt{\frac{a^{3}}{6}}$

$\frac{a \sqrt{6 a}}{6}$

$\sqrt{\frac{y^{4}}{x^{3}}}$

$\frac{y^{2} \sqrt{x}}{x^{2}}$

$\sqrt{\frac{32 a^{5}}{b^{7}}}$

$\frac{4 a^{2} \sqrt{2 a b}}{b^{4}}$

$\sqrt{{(x + 9)}^{2}}$

$x + 9$

$\sqrt{x^{2} + 14 x + 49}$

$x + 7$

Exercises

For the following problems, simplify each of the radical expressions.

$\sqrt{8 b^{2}}$

$2 b \sqrt{2}$

$\sqrt{20 a^{2}}$

$\sqrt{24 x^{4}}$

$2 x^{2} \sqrt{6}$

$\sqrt{27 y^{6}}$

$\sqrt{a^{5}}$

$a^{2} \sqrt{a}$

$\sqrt{m^{7}}$

$\sqrt{x^{11}}$

$x^{5} \sqrt{x}$

$\sqrt{y^{17}}$

$\sqrt{36 n^{9}}$

$6 n^{4} \sqrt{n}$

$\sqrt{49 x^{13}}$

$\sqrt{100 x^{5} y^{11}}$

$10 x^{2} y^{5} \sqrt{x y}$

$\sqrt{64 a^{7} b^{3}}$

$5 \sqrt{16 m^{6} n^{7}}$

$20 m^{3} n^{3} \sqrt{n}$

$8 \sqrt{9 a^{4} b^{11}}$

$3 \sqrt{16 x^{3}}$

$12 x \sqrt{x}$

$8 \sqrt{25 y^{3}}$

$\sqrt{12 a^{4}}$

$2 a^{2} \sqrt{3}$

$\sqrt{32 m^{8}}$

$\sqrt{32 x^{7}}$

$4 x^{3} \sqrt{2 x}$

$\sqrt{12 y^{13}}$

$\sqrt{50 a^{3} b^{9}}$

$5 a b^{4} \sqrt{2 a b}$

$\sqrt{48 p^{11} q^{5}}$

$4 \sqrt{18 a^{5} b^{17}}$

$12 a^{2} b^{8} \sqrt{2 a b}$

$8 \sqrt{108 x^{21} y^{3}}$

$- 4 \sqrt{75 a^{4} b^{6}}$

$- 20 a^{2} b^{3} \sqrt{3}$

$- 6 \sqrt{72 x^{2} y^{4} z^{10}}$

$- \sqrt{b^{12}}$

$- b^{6}$

$- \sqrt{c^{18}}$

$\sqrt{a^{2} b^{2} c^{2}}$

$a b c$

$\sqrt{4 x^{2} y^{2} z^{2}}$

$- \sqrt{9 a^{2} b^{3}}$

$- 3 a b \sqrt{b}$

$- \sqrt{16 x^{4} y^{5}}$

$\sqrt{m^{6} n^{8} p^{12} q^{20}}$

$m^{3} n^{4} p^{6} q^{10}$

$\sqrt{r^{2}}$

$\sqrt{p^{2}}$

$p$

$\sqrt{\frac{1}{4}}$

$\sqrt{\frac{1}{16}}$

$\frac{1}{4}$

$\sqrt{\frac{4}{25}}$

$\sqrt{\frac{9}{49}}$

$\frac{3}{7}$

$\frac{5 \sqrt{8}}{\sqrt{3}}$

$\frac{2 \sqrt{32}}{\sqrt{3}}$

$\frac{8 \sqrt{6}}{3}$

$\sqrt{\frac{5}{6}}$

$\sqrt{\frac{2}{7}}$

$\frac{\sqrt{14}}{7}$

$\sqrt{\frac{3}{10}}$

$\sqrt{\frac{4}{3}}$

$\frac{2 \sqrt{3}}{3}$

$- \sqrt{\frac{2}{5}}$

$- \sqrt{\frac{3}{10}}$

$- \frac{\sqrt{30}}{10}$

$\sqrt{\frac{16 a^{2}}{5}}$

$\sqrt{\frac{24 a^{5}}{7}}$

$\frac{2 a^{2} \sqrt{42 a}}{7}$

$\sqrt{\frac{72 x^{2} y^{3}}{5}}$

$\sqrt{\frac{2}{a}}$

$\frac{\sqrt{2 a}}{a}$

$\sqrt{\frac{5}{b}}$

$\sqrt{\frac{6}{x^{3}}}$

$\frac{\sqrt{6 x}}{x^{2}}$

$\sqrt{\frac{12}{y^{5}}}$

$\sqrt{\frac{49 x^{2} y^{5} z^{9}}{25 a^{3} b^{11}}}$

$\frac{7 x y^{2} z^{4} \sqrt{a b y z}}{5 a^{2} b^{6}}$

$\sqrt{\frac{27 x^{6} y^{15}}{3^{3} x^{3} y^{5}}}$

$\sqrt{{(b + 2)}^{4}}$

${(b + 2)}^{2}$

$\sqrt{{(a - 7)}^{8}}$

$\sqrt{{(x + 2)}^{6}}$

${(x + 2)}^{3}$

$\sqrt{{(x + 2)}^{2} {(x + 1)}^{2}}$

$\sqrt{{(a - 3)}^{4} {(a - 1)}^{2}}$

${(a - 3)}^{2} (a - 1)$

$\sqrt{{(b + 7)}^{8} {(b - 7)}^{6}}$

$\sqrt{a^{2} - 10 a + 25}$

$(a - 5)$

$\sqrt{b^{2} + 6 b + 9}$

$\sqrt{{(a^{2} - 2 a + 1)}^{4}}$

${(a - 1)}^{4}$

$\sqrt{{(x^{2} + 2 x + 1)}^{12}}$

Exercises for review

( [link] ) Solve the inequality $3 (a + 2) \leq 2 (3 a + 4)$

$a \geq - \frac{2}{3}$

( [link] ) Graph the inequality $6 x \leq 5 (x + 1) - 6.$

( [link] ) Supply the missing words. When looking at a graph from left-to-right, lines with _______ slope rise, while lines with __________ slope fall.

positive; negative

( [link] ) Simplify the complex fraction $\frac{5 + \frac{1}{x}}{5 - \frac{1}{x}} .$

( [link] ) Simplify $\sqrt{121 x^{4} w^{6} z^{8}}$ by removing the radical sign.

$11 x^{2} w^{3} z^{4}$

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Source: OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1

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