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:

Note

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

$\begin{array}{flushleft} \sqrt{9 \cdot 4} &= \sqrt{36} &= 6 & \text{ and }\\ \sqrt{9} \sqrt{4} &= 3 \cdot 2 &= 6 \end{array}$

The Product Property $\sqrt{xy} = \sqrt{x} \sqrt{y}$

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

$\sqrt{\dfrac{36}{4}} = \dfrac{\sqrt{36}}{\sqrt{4}}$

The Quotient Property $\sqrt{\dfrac{x}{y}} = \dfrac{\sqrt{x}}{\sqrt{y}}$

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.

CAUTION

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:

Simplifying Square Roots Without Fractions

1. If a factor of the radicand contains a variable with an even exponent, the square root is obtained by dividing the exponent by 2.
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.

Example $\PageIndex{1}$

$\sqrt{a^4}$. The exponent is even: $\dfrac{4}{2} = 2$. The exponent on the square root is $2$.

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

Example $\PageIndex{2}$

$\sqrt{a^6b^{10}}$. 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 $5$.

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

Example $\PageIndex{3}$

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

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

Example $\PageIndex{4}$

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

Example $\PageIndex{5}$

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

Example $\PageIndex{6}$

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

Practice Set A

Simplify each square root.

Practice Problem $\PageIndex{1}$

$\sqrt{m^8}$

Answer

$m^4$

Practice Problem $\PageIndex{2}$

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

Answer

$h^7k^{11}$

Practice Problem $\PageIndex{3}$

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

Answer

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

Practice Problem $\PageIndex{4}$

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

Answer

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

Practice Problem $\PageIndex{5}$

$\sqrt{w^5}$

Answer

$w^2 \sqrt{w}$

Practice Problem $\PageIndex{6}$

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

Answer

$w^3zk^6 \sqrt{wzk}$

Practice Problem $\PageIndex{7}$

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

Answer

$3ab^2c^2d^3 \sqrt{3ac}$

Practice Problem $\PageIndex{8}$

$\sqrt{180m^4n^{15}9a-12)^{15}}$

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

1. no perfect squares in the radicand,
2. no fractions in the radicand, or
3. 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.

Simplifying Square Roots with Fractions

To simplify the square root expression $\sqrt{\dfrac{x}{y}}$,

1. Write the expression as $\dfrac{\sqrt{x}}{\sqrt{y}}$ using the rule $\sqrt{\dfrac{x}{y}} = \dfrac{\sqrt{x}}{\sqrt{y}}$.
2. Multiply the fraction by 1 in the form of $\dfrac{\sqrt{y}}{\sqrt{y}}$.
3. 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.

Example $\PageIndex{7}$

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

Example $\PageIndex{8}$

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

Example $\PageIndex{9}$

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

Example $\PageIndex{10}$

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

Example $\PageIndex{11}$

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

Practice Set B

Simplify each square root.

Practice Problem $\PageIndex{9}$

$\sqrt{\dfrac{81}{25}}$

Answer

$\dfrac{9}{5}$

Practice Problem $\PageIndex{10}$

$\sqrt{\dfrac{2}{7}}$

Answer

$\dfrac{\sqrt{14}}{7}$

Practice Problem $\PageIndex{11}$

$\sqrt{\dfrac{4}{5}}$

Answer

$\dfrac{2 \sqrt{5}}{5}$

Practice Problem $\PageIndex{12}$

$\sqrt{\dfrac{10}{4}}$

Answer

$\dfrac{\sqrt{10}}{2}$

Practice Problem $\PageIndex{13}$

$\sqrt{\dfrac{9}{4}}$

Answer

$\dfrac{3}{2}$

Practice Problem $\PageIndex{14}$

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

Answer

$\dfrac{a \sqrt{6a}}{6}$

Practice Problem $\PageIndex{15}$

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

Answer

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

Practice Problem $\PageIndex{16}$

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

Answer

$\dfrac{4a^2 \sqrt{2ab}}{b^4}$

Practice Problem $\PageIndex{17}$

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

Answer

$x+9$

Practice Problem $\PageIndex{18}$

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

Answer

$x+7$

Exercises

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

Exercise $\PageIndex{1}$

$\sqrt{8b^2}$

Answer

$2b \sqrt{2}$

Exercise $\PageIndex{2}$

$\sqrt{20a^2}$

Exercise $\PageIndex{3}$

$\sqrt{24x^4}$

Answer

$2x^2 \sqrt{6}$

Exercise $\PageIndex{4}$

$\sqrt{27y^6}$

Exercise $\PageIndex{5}$

$\sqrt{a^5}$

Answer

$a^2\sqrt{a}$

Exercise $\PageIndex{6}$

$\sqrt{m^7}$

Exercise $\PageIndex{7}$

$\sqrt{x^{11}}$

Answer

$x^5 \sqrt{x}$

Exercise $\PageIndex{8}$

$\sqrt{y^{17}}$

Exercise $\PageIndex{9}$

$\sqrt{36n^9}$

Answer

$6n^4 \sqrt{n}$

Exercise $\PageIndex{10}$

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

Exercise $\PageIndex{11}$

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

Answer

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

Exercise $\PageIndex{12}$

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

Exercise $\PageIndex{13}$

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

Answer

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

Exercise $\PageIndex{14}$

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

Exercise $\PageIndex{15}$

$3 \sqrt{16x^3}$

Answer

$12x \sqrt{x}$

Exercise $\PageIndex{16}$

$8 \sqrt{25y^3}$

Exercise $\PageIndex{17}$

$\sqrt{12a^4}$

Answer

$2a^2 \sqrt{3}$

Exercise $\PageIndex{18}$

$\sqrt{32x^7}$

Answer

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

Exercise $\PageIndex{19}$

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

Exercise $\PageIndex{20}$

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

Answer

$5ab^4 \sqrt{2ab}$

Exercise $\PageIndex{21}$

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

Exercise $\PageIndex{22}$

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

Answer

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

Exercise $\PageIndex{23}$

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

Exercise $\PageIndex{24}$

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

Answer

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

Exercise $\PageIndex{25}$

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

Exercise $\PageIndex{26}$

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

Answer

$-b^6$

Exercise $\PageIndex{27}$

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

Exercise $\PageIndex{28}$

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

Answer

$abc$

Exercise $\PageIndex{29}$

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

Exercise $\PageIndex{30}$

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

Answer

$-3ab \sqrt{b}$

Exercise $\PageIndex{31}$

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

Exercise $\PageIndex{32}$

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

Answer

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

Exercise $\PageIndex{33}$

$\sqrt{r^2}$

Exercise $\PageIndex{34}$

$\sqrt{p^2}$

Answer

$p$

Exercise $\PageIndex{35}$

$\sqrt{\dfrac{1}{4}}$

Exercise $\PageIndex{36}$

$\sqrt{\dfrac{1}{16}}$

Answer

$\dfrac{1}{4}$

Exercise $\PageIndex{37}$

$\sqrt{\dfrac{4}{25}}$

Exercise $\PageIndex{38}$

$\sqrt{\dfrac{9}{49}}$

Answer

$\dfrac{3}{7}$

Exercise $\PageIndex{39}$

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

Exercise $\PageIndex{40}$

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

Answer

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

Exercise $\PageIndex{41}$

$\sqrt{\dfrac{5}{6}}$

Exercise $\PageIndex{42}$

$\sqrt{\dfrac{2}{7}}$

Answer

$\dfrac{\sqrt{14}}{7}$

Exercise $\PageIndex{43}$

$\sqrt{\dfrac{3}{10}}$

Exercise $\PageIndex{44}$

$\sqrt{\dfrac{4}{3}}$

Answer

$\dfrac{2 \sqrt{3}}{3}$

Exercise $\PageIndex{45}$

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

Exercise $\PageIndex{46}$

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

Answer

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

Exercise $\PageIndex{47}$

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

Exercise $\PageIndex{48}$

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

Answer

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

Exercise $\PageIndex{49}$

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

Exercise $\PageIndex{50}$

$\sqrt{\dfrac{2}{a}}$

Answer

$\dfrac{\sqrt{2a}}{a}$

Exercise $\PageIndex{51}$

$\sqrt{\dfrac{5}{b}}$

Exercise $\PageIndex{52}$

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

Answer

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

Exercise $\PageIndex{53}$

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

Exercise $\PageIndex{54}$

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

Answer

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

Exercise $\PageIndex{55}$

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

Exercise $\PageIndex{56}$

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

Answer

$(b+2)^2$

Exercise $\PageIndex{57}$

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

Exercise $\PageIndex{58}$

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

Answer

$(x+2)^3$

Exercise $\PageIndex{59}$

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

Exercise $\PageIndex{60}$

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

Answer

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

Exercise $\PageIndex{61}$

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

Exercise $\PageIndex{62}$

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

Answer

$(a-5)$

Exercise $\PageIndex{63}$

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

Exercise $\PageIndex{64}$

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

Answer

$(a-1)^4$

Exercise $\PageIndex{65}$

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

Exercises For Review

Exercise $\PageIndex{66}$

Solve the inequality $3(a + 2) \le 2(3a + 4)$

Answer

$a \ge -\dfrac{2}{3}$

Exercise $\PageIndex{67}$

Graph the inequality $6x \le 5(x+1) - 6$

Exercise $\PageIndex{68}$

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

Exercise $\PageIndex{69}$

Simplify the complex fraction: $\dfrac{5+\frac{1}{x}}{5-\frac{1}{x}}$

Exercise $\PageIndex{70}$

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

Answer

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