# 9.5: Division of Square Root Expressions

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## The Division Property of Square Roots

In our work with simplifying square root expressions, we noted that

$\sqrt{\dfrac{x}{y}} = \dfrac{\sqrt{x}}{\sqrt{y}} \nonumber$

Since this is an equation, we may write it as:

$\dfrac{\sqrt{x}}{\sqrt{y}} = \sqrt{\dfrac{x}{y}} \nonumber$

To divide two square root expressions, we use the division property of square roots:

##### The Division Property: $$\dfrac{\sqrt{x}}{\sqrt{y}} = \sqrt{\dfrac{x}{y}}$$

$\dfrac{\sqrt{x}}{\sqrt{y}} = \sqrt{\dfrac{x}{y}}$

The quotient of the square root is the square root of the quotient.

## Rationalizing the Denominator

As we can see by observing the right side of the equation governing the division of square roots, the process may produce a fraction in the radicand. This means, of course, that the square root expression is not in simplified form. It is sometimes more useful to rationalize the denominator of a square root expression before actually performing the division.

### Sample Set A

Simplify the square root expressions.

##### Example $$\PageIndex{1}$$

$\sqrt{\dfrac{3}{7}} \nonumber$

This radical expression is not in simplified form since there is a fraction under the radical sign. We can eliminate this problem using the division property of square roots.

$\sqrt{\dfrac{3}{7}} = \dfrac{\sqrt{3}}{\sqrt{7}} = \dfrac{\sqrt{3}}{\sqrt{7}} \cdot \dfrac{\sqrt{7}}{\sqrt{7}} = \dfrac{\sqrt{3} \sqrt{7}}{7} = \dfrac{\sqrt{21}}{7} \nonumber$

##### Example $$\PageIndex{2}$$

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

A direct application of the rule produces $$\sqrt{\dfrac{5}{3}}$$, which must be simplified. Let us rationalize the denominator before we perform the division.

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

##### Example $$\PageIndex{3}$$

$$\dfrac{\sqrt{21}}{\sqrt{7}} = \sqrt{\dfrac{21}{7}} = \sqrt{3}$$

The rule produces the quotient quickly. We could also rationalize the denominator first and produce the same result.

$$\dfrac{\sqrt{21}}{\sqrt{7}}=\dfrac{\sqrt{21}}{7} \cdot \dfrac{\sqrt{7}}{\sqrt{7}}=\dfrac{\sqrt{21 \cdot 7}}{7}=\dfrac{\sqrt{3 \cdot 7 \cdot 7}}{7}=\dfrac{\sqrt{3 \cdot 7^{2}}}{7}=\dfrac{7 \sqrt{3}}{7}=\sqrt{3}$$

##### Example $$\PageIndex{4}$$

$$\dfrac{\sqrt{80 x^{9}}}{\sqrt{5 x^{4}}}=\sqrt{\dfrac{80 x^{9}}{5 x^{4}}}=\sqrt{16 x^{5}}=\sqrt{16} \sqrt{x^{4} x}=4 x^{2} \sqrt{x}$$

##### Example $$\PageIndex{5}$$

$$\dfrac{\sqrt{50 a^{3} b^{7}}}{\sqrt{5 a b^{5}}}=\sqrt{\dfrac{50 a^{3} b^{7}}{5 a b^{5}}}=\sqrt{10 a^{2} b^{2}}=a b \sqrt{10}$$

##### Example $$\PageIndex{6}$$

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

Some observation shows that a direct division of the radicands will produce a fraction. This suggests that we rationalize the denominator first.

$$\dfrac{\sqrt{5a}}{\sqrt{b}} = \dfrac{\sqrt{5a}}{\sqrt{b}} \cdot \dfrac{\sqrt{b}}{\sqrt{b}} = \dfrac{\sqrt{5a} \sqrt{b}}{b} = \dfrac{\sqrt{5ab}}{b}$$

##### Example $$\PageIndex{7}$$

$$\dfrac{\sqrt{m-6}}{\sqrt{m+2}} = \dfrac{\sqrt{m-6}}{\sqrt{m+2}} \cdot \dfrac{\sqrt{m+2}}{\sqrt{m+2}} = \dfrac{\sqrt{m^2 - 4m - 12}}{m + 2}$$

##### Example $$\PageIndex{8}$$

$$\dfrac{\sqrt{y^{2}-y-12}}{\sqrt{y+3}}=\sqrt{\dfrac{y^{2}-y-12}{y+3}}=\sqrt{\dfrac{(y+3)(y-4)}{(y+3)}}=\sqrt{\dfrac{\cancel{(y+3)}(y-4)}{\cancel{(y+3)}}}=\sqrt{y-4}$$

### Practice Set A

Simplify the square root expressions.

##### Practice Problem $$\PageIndex{1}$$

$$\dfrac{\sqrt{26}}{\sqrt{13}}$$

$$\sqrt{2}$$

##### Practice Problem $$\PageIndex{2}$$

$$\dfrac{\sqrt{7}}{\sqrt{3}}$$

$$\dfrac{\sqrt{21}}{3}$$

##### Practice Problem $$\PageIndex{3}$$

$$\dfrac{\sqrt{80m^5n^8}}{\sqrt{5m^2n}}$$

$$4mn^3 \sqrt{mn}$$

##### Practice Problem $$\PageIndex{4}$$

$$\dfrac{\sqrt{196(x+7)^8}}{\sqrt{2(x+7)^3}}$$

$$7(x+7)^2 \sqrt{2(x+7)}$$

##### Practice Problem $$\PageIndex{5}$$

$$\dfrac{\sqrt{n+4}}{\sqrt{n-5}}$$

$$\dfrac{\sqrt{n^2 - n - 20}}{n-5}$$

##### Practice Problem $$\PageIndex{6}$$

$$\dfrac{\sqrt{a^2 - 6a + 8}}{\sqrt{a-2}}$$

$$\sqrt{a-4}$$

##### Practice Problem $$\PageIndex{7}$$

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

$$x^n$$

##### Practice Problem $$\PageIndex{8}$$

$$\dfrac{\sqrt{a^{3m-5}}}{\sqrt{a^{m-1}}}$$

$$a^{m-2}$$

## Conjugates and Rationalizing the Denominator

To perform a division that contains a binomial in the denominator, such as $$\dfrac{3}{4 + \sqrt{6}}$$, we multiply the numerator and denominator by a conjugate of the denominator.

##### Conjugate

A conjugate of the binomial $$a + b$$ is $$a-b$$. Similarly, a conjugate of $$a-b$$ is $$a + b$$.

Notice that when the conjugates $$a + b$$ and $$a - b$$ are multiplied together, they produce a difference of two squares.

$$(a+b)(a-b) = a^2 - ab + ab - b^2 = a^2 - b^2$$

This principle helps us eliminate square root radicals, as shown in these examples that illustrate the produce of conjugates.

##### Example $$\PageIndex{9}$$

$$\begin{array}{flushleft} (5 + \sqrt{2})(5 - \sqrt{2}) &= 5^2 - (\sqrt{2})^2\\ &= 25 - 2\\ &= 23 \end{array}$$

##### Example $$\PageIndex{10}$$

$$\begin{array}{flushleft} (\sqrt{6} - \sqrt{7})(\sqrt{6} + \sqrt{7}) &= (\sqrt{6})^2 - (\sqrt{7})^2\\ &= 6 - 7\\ &= -1 \end{array}$$

## Sample Set B

Simplify the following expressions.

##### Example $$\PageIndex{11}$$

$$\dfrac{3}{4 + \sqrt{6}}$$

The conjugate of the denominator is $$4 - \sqrt{6}$$. Multiply the fraction by $$1$$ in the form of $$\dfrac{4 - \sqrt{6}}{4 - \sqrt{6}}$$

$$\begin{array}{flushleft} \dfrac{3}{4 + \sqrt{6}} \cdot \dfrac{4 - \sqrt{6}}{4 - \sqrt{6}} &= \dfrac{3(4 - \sqrt{6})}{4^2 - (\sqrt{6})^2}\\ &= \dfrac{12 - 3\sqrt{6}}{16 - 6}\\ &= \dfrac{12 - 3\sqrt{6}}{10} \end{array}$$

##### Example $$\PageIndex{12}$$

$$\dfrac{\sqrt{2x}}{\sqrt{3} - \sqrt{5x}}$$

The conjugate of the denominator is $$\sqrt{3} + \sqrt{5x}$$. Multiply the fraction by $$1$$ in the form of $$\dfrac{\sqrt{3} + \sqrt{5x}}{\sqrt{3} + \sqrt{5x}}$$.

$$\begin{array}{flushleft} \dfrac{\sqrt{2 x}}{\sqrt{3}-\sqrt{5 x}} \cdot \dfrac{\sqrt{3}+\sqrt{5 x}}{\sqrt{3}+\sqrt{5 x}} &=\dfrac{\sqrt{2 x}(\sqrt{3}+\sqrt{5 x})}{(\sqrt{3})^{2}-(\sqrt{5 x})^{2}} \\ &=\dfrac{\sqrt{2 x} \sqrt{3}+\sqrt{2 x} \sqrt{5 x}}{3-5 x} \\ &=\dfrac{\sqrt{6 x}+\sqrt{10 x^{2}}}{3-5 x} \\ &=\dfrac{\sqrt{6 x}+x \sqrt{10}}{3-5 x} \end{array}$$

## Practice Set B

Simplify the following expressions.

##### Practice Problem $$\PageIndex{9}$$

$$\dfrac{5}{9 + \sqrt{7}}$$

$$\dfrac{45 - 5\sqrt{7}}{74}$$

##### Practice Problem $$\PageIndex{10}$$

$$\dfrac{-2}{1 - \sqrt{3x}}$$

$$\dfrac{-2 - 2\sqrt{3x}}{1 - 3x}$$

##### Practice Problem $$\PageIndex{11}$$

$$\dfrac{\sqrt{8}}{\sqrt{3x} + \sqrt{2x}}$$

$$\dfrac{2\sqrt{6x} - 4\sqrt{x}}{x}$$

##### Practice Problem $$\PageIndex{12}$$

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

$$\dfrac{\sqrt{2m} + \sqrt{6}}{m - 3}$$

## Exercises

For the following problems, simplify each expression.

##### Exercise $$\PageIndex{1}$$

$$\dfrac{\sqrt{28}}{\sqrt{2}}$$

$$\sqrt{14}$$

##### Exercise $$\PageIndex{2}$$

$$\dfrac{\sqrt{200}}{\sqrt{10}}$$

##### Exercise $$\PageIndex{3}$$

$$\dfrac{\sqrt{28}}{\sqrt{7}}$$

$$2$$

##### Exercise $$\PageIndex{4}$$

$$\dfrac{\sqrt{96}}{\sqrt{24}}$$

##### Exercise $$\PageIndex{5}$$

$$\dfrac{\sqrt{180}}{\sqrt{5}}$$

$$6$$

##### Exercise $$\PageIndex{6}$$

$$\dfrac{\sqrt{336}}{\sqrt{21}}$$

##### Exercise $$\PageIndex{7}$$

$$\dfrac{\sqrt{162}}{\sqrt{18}}$$

$$3$$

##### Exercise $$\PageIndex{8}$$

$$\sqrt{\dfrac{25}{9}}$$

##### Exercise $$\PageIndex{9}$$

$$\sqrt{\dfrac{36}{35}}$$

$$\dfrac{6\sqrt{35}}{35}$$

##### Exercise $$\PageIndex{10}$$

$$\sqrt{\dfrac{225}{16}}$$

##### Exercise $$\PageIndex{11}$$

$$\sqrt{\dfrac{49}{225}}$$

$$\dfrac{7}{15}$$

##### Exercise $$\PageIndex{12}$$

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

##### Exercise $$\PageIndex{13}$$

$$\sqrt{\dfrac{3}{7}}$$

$$\dfrac{\sqrt{21}}{7}$$

##### Exercise $$\PageIndex{14}$$

$$\sqrt{\dfrac{1}{2}}$$

##### Exercise $$\PageIndex{15}$$

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

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

##### Exercise $$\PageIndex{16}$$

$$\sqrt{\dfrac{11}{25}}$$

##### Exercise $$\PageIndex{17}$$

$$\sqrt{\dfrac{15}{36}}$$

$$\dfrac{\sqrt{15}}{6}$$

##### Exercise $$\PageIndex{18}$$

$$\sqrt{\dfrac{5}{16}}$$

##### Exercise $$\PageIndex{19}$$

$$\sqrt{\dfrac{7}{25}}$$

$$\dfrac{\sqrt{7}}{5}$$

##### Exercise $$\PageIndex{20}$$

$$\sqrt{\dfrac{32}{49}}$$

##### Exercise $$\PageIndex{21}$$

$$\sqrt{\dfrac{50}{81}}$$

$$\dfrac{5 \sqrt{2}}{9}$$

##### Exercise $$\PageIndex{22}$$

$$\dfrac{\sqrt{125x^5}}{\sqrt{5x^3}}$$

##### Exercise $$\PageIndex{23}$$

$$\dfrac{\sqrt{72m^7}}{\sqrt{2m^3}}$$

$$6m^2$$

##### Exercise $$\PageIndex{24}$$

$$\dfrac{\sqrt{162a^{11}}}{\sqrt{2a^5}}$$

##### Exercise $$\PageIndex{25}$$

$$\dfrac{\sqrt{75y^{10}}}{\sqrt{2a^5}}$$

$$5y^3$$

##### Exercise $$\PageIndex{26}$$

$$\dfrac{\sqrt{48x^9}}{\sqrt{3x^2}}$$

##### Exercise $$\PageIndex{27}$$

$$\dfrac{\sqrt{125a^{14}}}{\sqrt{5a^5}}$$

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

##### Exercise $$\PageIndex{28}$$

$$\dfrac{\sqrt{27a^{10}}}{\sqrt{3a^5}}$$

##### Exercise $$\PageIndex{29}$$

$$\dfrac{\sqrt{108x^{21}}}{\sqrt{3x^4}}$$

$$6x^8 \sqrt{x}$$

##### Exercise $$\PageIndex{30}$$

$$\dfrac{\sqrt{48x^6y^7}}{\sqrt{3xy}}$$

##### Exercise $$\PageIndex{31}$$

$$\dfrac{\sqrt{45a^3b^8c^2}}{\sqrt{5ab^2c}}$$

$$3ab^3 \sqrt{c}$$

##### Exercise $$\PageIndex{32}$$

$$\dfrac{\sqrt{66m^{12}n^{15}}}{\sqrt{11mn^8}}$$

##### Exercise $$\PageIndex{33}$$

$$\dfrac{\sqrt{30p^5q^{14}}}{\sqrt{5q^7}}$$

$$p^2q^3 \sqrt{6pq}$$

##### Exercise $$\PageIndex{34}$$

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

##### Exercise $$\PageIndex{35}$$

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

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

##### Exercise $$\PageIndex{36}$$

$$\dfrac{\sqrt{2a^3b}}{\sqrt{14a}}$$

##### Exercise $$\PageIndex{37}$$

$$\dfrac{\sqrt{3m^4n^3}}{\sqrt{6mn^5}}$$

$$\dfrac{m \sqrt{2m}}{2n}$$

##### Exercise $$\PageIndex{38}$$

$$\dfrac{\sqrt{5(p-q)^6(r+s)^4}}{\sqrt{25(r+s)^3}}$$

##### Exercise $$\PageIndex{39}$$

$$\dfrac{\sqrt{m(m-6)-m^2 + 6m}}{\sqrt{3m - 7}}$$

$$0$$

##### Exercise $$\PageIndex{40}$$

$$\dfrac{\sqrt{r+1}}{\sqrt{r-1}}$$

##### Exercise $$\PageIndex{41}$$

$$\dfrac{\sqrt{s+3}}{\sqrt{s-3}}$$

$$\dfrac{\sqrt{s^2-9}}{s-3}$$

##### Exercise $$\PageIndex{42}$$

$$\dfrac{\sqrt{a^2 + 3a + 2}}{\sqrt{a + 1}}$$

##### Exercise $$\PageIndex{43}$$

$$\dfrac{\sqrt{x^2 - 10x + 24}}{\sqrt{x-4}}$$

$$\sqrt{x-6}$$

##### Exercise $$\PageIndex{44}$$

$$\dfrac{\sqrt{x^2 - 2x - 8}}{\sqrt{x + 2}}$$

##### Exercise $$\PageIndex{45}$$

$$\dfrac{\sqrt{x^2 - 4x + 3}}{\sqrt{x-3}}$$

$$\sqrt{x-1}$$

##### Exercise $$\PageIndex{46}$$

$$\dfrac{\sqrt{2x^2 - x - 1}}{\sqrt{x - 1}}$$

##### Exercise $$\PageIndex{47}$$

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

$$\dfrac{-20 + 5\sqrt{5}}{11}$$

##### Exercise $$\PageIndex{48}$$

$$\dfrac{1}{1 + \sqrt{x}}$$

##### Exercise $$\PageIndex{49}$$

$$\dfrac{2}{1 - \sqrt{a}}$$

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

##### Exercise $$\PageIndex{50}$$

$$\dfrac{-6}{\sqrt{5} - 1}$$

##### Exercise $$\PageIndex{51}$$

$$\dfrac{-6}{\sqrt{7} + 2}$$

$$-2(\sqrt{7} - 2)$$

##### Exercise $$\PageIndex{52}$$

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

##### Exercise $$\PageIndex{53}$$

$$\dfrac{4}{\sqrt{6} + \sqrt{2}}$$

$$\sqrt{6} - \sqrt{2}$$

##### Exercise $$\PageIndex{54}$$

$$\dfrac{\sqrt{5}}{\sqrt{8} - \sqrt{6}}$$

##### Exercise $$\PageIndex{55}$$

$$\dfrac{\sqrt{12}}{\sqrt{12} - \sqrt{8}}$$

$$3 + \sqrt{6}$$

##### Exercise $$\PageIndex{56}$$

$$\dfrac{\sqrt{7x}}{2 - \sqrt{5x}}$$

##### Exercise $$\PageIndex{57}$$

$$\dfrac{\sqrt{6y}}{1 + \sqrt{3y}}$$

$$\dfrac{\sqrt{6y} - 3y\sqrt{2}}{1 - 3y}$$

##### Exercise $$\PageIndex{58}$$

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

##### Exercise $$\PageIndex{59}$$

$$\dfrac{\sqrt{a}}{\sqrt{a} + \sqrt{b}}$$

$$\dfrac{a - \sqrt{ab}}{a - b}$$

##### Exercise $$\PageIndex{60}$$

$$\dfrac{\sqrt{8^3b^5}}{4 - \sqrt{2ab}}$$

##### Exercise $$\PageIndex{61}$$

$$\dfrac{\sqrt{7x}}{\sqrt{5x} + \sqrt{x}}$$

$$\dfrac{\sqrt{35} - \sqrt{7}}{4}$$

##### Exercise $$\PageIndex{62}$$

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

## Exercises for Review

##### Exercise $$\PageIndex{63}$$

Simplify $$x^8y^7 \dfrac{x^4y^8}{x^3y^4}$$

$$x^9y^{11}$$

##### Exercise $$\PageIndex{64}$$

Solve the compound inequality $$-8 \le 7 - 5x \le -23$$

##### Exercise $$\PageIndex{65}$$

Construct the graph of $$y = \dfrac{2}{3}x - 4$$  ##### Exercise $$\PageIndex{66}$$

The symbol $$\sqrt{x}$$ represents which square root of the number $$x, x \ge 0$$?

##### Exercise $$\PageIndex{67}$$

Simplify $$\sqrt{a^2 + 8a + 16}$$

$$a + 4$$