Divide Square Roots

We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.

This figure shows two columns. The first is labeled “Common Factors” and has 3 times the square root of 2 over 3 times 5 beneath it. Both number threes are red. The second column is labeled “No common factors” and has 2 times the square root of 3 over 3 times 5.

Simplify: $\frac{\sqrt{54}}{6}$ .

Solution

This table illustrates the step-by-step simplification of the radical expression sqrt(54)/6, demonstrating the process from its initial to its final simplified form.
	$\frac{\sqrt{54}}{6}$
Simplify the radical.	$\frac{\sqrt{9} \cdot \sqrt{6}}{6}$
Simplify.	$\frac{3 \sqrt{6}}{6}$
Remove the common factors.	$\frac{3 \sqrt{6}}{3 \cdot 2}$
Simplify.	$\frac{\sqrt{6}}{2}$

Simplify: $\frac{6 - \sqrt{24}}{12}$ .

Solution

Step-by-step guide to simplifying the radical expression (6 - sqrt(24)) / 12, demonstrating each transformation to arrive at (3 - sqrt(6)) / 6.
	$\frac{6 - \sqrt{24}}{12}$
Simplify the radical.	$\frac{6 - \sqrt{4} \cdot \sqrt{6}}{12}$
Simplify.	$\frac{6 - 2 \sqrt{6}}{12}$
Factor the common factor from the numerator.	$\frac{2 (3 - \sqrt{6})}{2 \cdot 6}$
Remove the common factors.	$\frac{2 (3 - \sqrt{6})}{2 \cdot 6}$
Simplify.	$\frac{3 - \sqrt{6}}{6}$

We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says

\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}, b \neq 0

Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.

\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}, b \neq 0

We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.

We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.

Simplify: $\frac{\sqrt{27}}{\sqrt{75}}$ .

Solution

Step-by-step guide on simplifying the rational expression sqrt(27)/sqrt(75), detailing each procedural step and the corresponding mathematical expression.
	$\frac{\sqrt{27}}{\sqrt{75}}$
Neither radicand is a perfect square, so rewrite using the quotient property of square roots.	$\sqrt{\frac{27}{75}}$
Remove common factors in the numerator and denominator.	$\sqrt{\frac{3 \cdot 9}{3 \cdot 25}}$
Simplify.	$\sqrt{\frac{9}{25}}$
	$\frac{3}{5}$

We will use the Quotient Property for Exponents, $\frac{a^{m}}{a^{n}} = a^{m - n}$ , when we have variables with exponents in the radicands.

Simplify: $\frac{\sqrt{6 y^{5}}}{\sqrt{2 y}}$ .

Solution

Steps to simplify the quotient of two square root expressions involving variables.
	$\frac{\sqrt{6 y^{5}}}{\sqrt{2 y}}$
Neither radicand is a perfect square, so rewrite using the quotient property of square roots.	$\sqrt{\frac{6 y^{5}}{2 y}}$
Remove common factors in the numerator and denominator.	$\sqrt{\frac{2 \cdot 3 \cdot y^{4} \cdot y}{2 \cdot y}}$
Simplify.	$\sqrt{3 y^{4}}$
Simplify the radical.	$y^{2} \sqrt{3}$

Simplify: $\frac{\sqrt{72 x^{3}}}{\sqrt{162 x}}$ .

Solution

This table illustrates the step-by-step simplification of a radical expression involving a quotient of square roots.
	$\frac{\sqrt{72 x^{3}}}{\sqrt{162 x}}$
Rewrite using the quotient property of square roots.	$\sqrt{\frac{72 x^{3}}{162 x}}$
Remove common factors.	$\sqrt{\frac{18 \cdot 4 \cdot x^{2} \cdot x}{18 \cdot 9 \cdot x}}$
Simplify.	$\sqrt{\frac{4 x^{2}}{9}}$
Simplify the radical.	$\frac{2 x}{3}$

Simplify: $\frac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}$ .

Solution

Step-by-step simplification of a rational expression with square roots, demonstrating algebraic properties.
	$\frac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}$
Rewrite using the quotient property of square roots.	$\sqrt{\frac{147 a b^{8}}{3 a^{3} b^{4}}}$
Remove common factors.	$\sqrt{\frac{49 b^{4}}{a^{2}}}$
Simplify the radical.	$\frac{7 b^{2}}{a}$

Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. Figure 1 shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.

This table has three solumn and eleven rows. The columns are labeled, “n,” “n squared,” and “the square root of n.” Under the column labeled “n” are the following numbers: 200; 201; 202; 203; 204; 205; 206; 207; 208; 209; and 210. Under the column labeled, “n squared” are the following numbers: 40,000; 40,401; 40,804; 41,209; 41,616; 42,025; 42,436; 42,849; 43,264; 43,681; 44,100. Under the column labeled, “the square root of n” are the following numbers: 14.14214; 14.17745; 14.21267; 14.24781; 14.28286; 14.31782; 14.35270; 14.38749; 14.42221; 14.45683; 14.49138. — A table of square roots was used to find approximate values of square roots before there were calculators.

If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimal-place divisor. This was a very cumbersome process.

For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.

Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.

Let’s look at a numerical example.

$\begin{matrix} Suppose we need an approximate value for the fraction. & \frac{1}{\sqrt{2}} \\ A five decimal place approximation to \sqrt{2} is 1.41421 . & \frac{1}{1.41421} \\ Without a calculator, would you want to do this division? & 1.41421 1.0 \end{matrix}$

But we can find a fraction equivalent to $\frac{1}{\sqrt{2}}$ by multiplying the numerator and denominator by $\sqrt{2}$ .

$This figure shows three fractions. The first fraction is 1 over the square root of 2. The second is 1 times the square root of 2 over the square root of 2 times the square root of 2. The third shows the square root of 2 over 2.$

Now if we need an approximate value, we divide $2 1.41421$ . This is much easier.

Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.

Similarly, a square root is not considered simplified if the radicand contains a fraction.

To rationalize a denominator, we use the property that ${(\sqrt{a})}^{2} = a$ . If we square an irrational square root, we get a rational number.

We will use this property to rationalize the denominator in the next example.

Simplify: $\frac{4}{\sqrt{3}}$ .

Solution

To rationalize a denominator, we can multiply a square root by itself. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

This table illustrates the step-by-step process of rationalizing the denominator of a fraction containing a square root.
	$\frac{4}{\sqrt{3}}$
Multiply both the numerator and denominator by $\sqrt{3} .$	$\frac{4 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}$
Simplify.	$\frac{4 \sqrt{3}}{3}$

Simplify: $- \frac{8}{3 \sqrt{6}}$ .

Solution

To remove the square root from the denominator, we multiply it by itself. To keep the fractions equivalent, we multiply both the numerator and denominator by $\sqrt{6}$ .

	$A mathematical expression displaying a fraction with a negative sign, where the numerator is 8 and the denominator is 3 multiplied by the square root of 6, represented as -8 / (3sqrt(6)).$
Multiply both the numerator and the denominator by $\sqrt{6}$ .
Simplify.	$A mathematical expression showing a negative fraction: - (8 times the square root of 6) divided by (3 multiplied by 6).$
Remove common factors.	$A mathematical expression showing a negative fraction where the factor 2 is crossed out in both the numerator and denominator. The numerator is 4 times the cancelled 2 times the square root of 6, and the denominator is 3 times the cancelled 2 times 3.$
Simplify.

Always simplify the radical in the denominator first, before you rationalize it. This way the numbers stay smaller and easier to work with.

Simplify: $\sqrt{\frac{5}{12}}$ .

Solution

	$A mathematical expression displays the square root of the fraction 5/12, with the numerator 5 and the denominator 12 clearly visible under the radical symbol.$
The fraction is not a perfect square, so rewrite using the Quotient Property.
Simplify the denominator	$A mathematical fraction with the square root of 5 in the numerator and 2 times the square root of 3 in the denominator.$
Rationalize the denominator.
Simplify.	$A mathematical fraction with the square root of 15 in the numerator and 2 multiplied by 3 in the denominator.$
Simplify.

Simplify: $\sqrt{\frac{11}{28}}$ .

Solution

	$A mathematical expression showing the square root of the fraction 11 over 28.$
Rewrite using the Quotient Property.	$A mathematical expression showing the square root of 11 divided by the square root of 28, presented as a fraction.$
Simplify the denominator.
Rationalize the denominator.
Simplify.
Simplify.

Rationalize a Two-Term Denominator

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.

\begin{matrix} (a - b) (a + b) & (2 - \sqrt{5}) (2 + \sqrt{5}) \\ a^{2} - b^{2} & 2^{2} - {(\sqrt{5})}^{2} \\ 4 - 5 \\ −1 \end{matrix}

When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.

Simplify: $\frac{4}{4 + \sqrt{2}}$ .

Solution

	$A fraction with 4 in the numerator and 4 + square root of 2 in the denominator.$
Multiply the numerator and denominator by the conjugate of the denominator.	$A mathematical expression showing the fraction 4(4-sqrt(2)) divided by the product of (4+sqrt(2)) and (4-sqrt(2)), with (4-sqrt(2)) highlighted in red in both the numerator and denominator.$
Multiply the conjugates in the denominator.	$A mathematical fraction is shown. The numerator is 4(4 - square root of 2). The denominator is 4 squared - (square root of 2) squared.$
Simplify the denominator.	$A mathematical expression showing a fraction. The numerator is 4(4 - square root of 2), and the denominator is 16 - 2.$
Simplify the denominator.	$A mathematical expression showing the fraction 4 multiplied by (4 minus the square root of 2), all divided by 14.$
Remove common factors from the numerator and denominator.	$A mathematical expression representing the fraction 2 multiplied by the quantity 4 minus the square root of 2, all divided by 7.$
We leave the numerator in factored form to make it easier to look for common factors after we have simplified the denominator.

Simplify: $\frac{5}{2 - \sqrt{3}}$ .

Solution

	$A mathematical fraction is displayed, with 5 as the numerator and (2 - sqrt(3)) as the denominator. This represents the expression 5 / (2 minus the square root of 3).$
Multiply the numerator and denominator by the conjugate of the denominator.	$A fraction: 5(2 + sqrt(3)) / ((2 - sqrt(3))(2 + sqrt(3))). The common term (2 + sqrt(3)) is highlighted in red in both the numerator and denominator, indicating a step in simplification.$
Multiply the conjugates in the denominator.	$A mathematical expression showing a fraction. The numerator is 5 times (2 plus the square root of 3). The denominator is 2 squared minus the square root of 3 squared.$
Simplify the denominator.	$A mathematical expression showing a fraction with 5 multiplied by (2 plus the square root of 3) in the numerator, and (4 minus 3) in the denominator, on a white background.$
Simplify the denominator.	$A mathematical expression showing the fraction 5(2 + 3)/1.$
Simplify.

Simplify: $\frac{\sqrt{3}}{\sqrt{u} - \sqrt{6}}$ .

Solution

	$A mathematical expression showing a fraction with the square root of 3 in the numerator and the square root of u minus the square root of 6 in the denominator.$
Multiply the numerator and denominator by the conjugate of the denominator.	$A fraction with a square root of three times the quantity of the square root of u plus the square root of six in the numerator. The denominator is the product of the quantity of the square root of u minus the square root of six and the quantity of the square root of u plus the square root of six.$
Multiply the conjugates in the denominator.	$A mathematical expression showing the fraction: square root of 3 multiplied by the sum of square root of 'u' and square root of 6, all divided by 'u' minus 6.$
Simplify the denominator.	$A fraction with a numerator of 'square root 3 times the quantity square root u plus square root 6' and a denominator of 'u minus 6'.$

Simplify: $\frac{\sqrt{x} + \sqrt{7}}{\sqrt{x} - \sqrt{7}}$ .

Solution

	$A mathematical fraction is displayed, with the numerator as the square root of x plus the square root of 7, and the denominator as the square root of x minus the square root of 7.$
Multiply the numerator and denominator by the conjugate of the denominator.	$An algebraic fraction showing (sqrt(x)+sqrt(7))^2 divided by (sqrt(x)-sqrt(7))(sqrt(x)+sqrt(7)), illustrating a step in simplifying radical expressions.$
Multiply the conjugates in the denominator.	$A mathematical fraction with the numerator as the product of (square root x + square root 7) and (square root x + square root 7), and the denominator as (square root x) squared minus (square root 7) squared.$
Simplify the denominator.	$A fraction with numerator (sqrt(x) + sqrt(7))^2 and denominator (x - 7).$
We do not square the numerator. In factored form, we can see there are no common factors to remove from the numerator and denominator.

Key Concepts

Quotient Property of Square Roots
- If a, b are non-negative real numbers and $b \neq 0$ , then
  $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} and \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
Simplified Square Roots
A square root is considered simplified if there are
- no perfect square factors in the radicand
- no fractions in the radicand
- no square roots in the denominator of a fraction

Practice Makes Perfect

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{27}}{6}$

Solution

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

$\frac{\sqrt{50}}{10}$

$\frac{\sqrt{72}}{9}$

Solution

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

$\frac{\sqrt{243}}{6}$

$\frac{2 - \sqrt{32}}{8}$

Solution

$\frac{1 - 2 \sqrt{2}}{4}$

$\frac{3 + \sqrt{27}}{9}$

$\frac{6 + \sqrt{45}}{6}$

Solution

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

$\frac{10 - \sqrt{200}}{20}$

$\frac{\sqrt{80}}{\sqrt{125}}$

Solution

$\frac{4}{5}$

$\frac{\sqrt{72}}{\sqrt{200}}$

$\frac{\sqrt{128}}{\sqrt{72}}$

Solution

$\frac{4}{3}$

$\frac{\sqrt{48}}{\sqrt{75}}$

ⓐ $\frac{\sqrt{8 x^{6}}}{\sqrt{2 x^{2}}}$ ⓑ $\frac{\sqrt{200 m^{5}}}{\sqrt{98 m}}$

Solution

ⓐ $2 x^{2}$ ⓑ $\frac{10 m^{2}}{7}$

ⓐ $\frac{\sqrt{10 y^{3}}}{\sqrt{5 y}}$ ⓑ $\frac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}$

$\frac{\sqrt{75 r^{3}}}{\sqrt{108 r}}$

Solution

$\frac{5 r}{6}$

$\frac{\sqrt{196 q^{5}}}{\sqrt{484 q}}$

$\frac{\sqrt{108 p^{5} q^{2}}}{\sqrt{3 p^{3} q^{6}}}$

Solution

$\frac{6 p \sqrt{102}}{q^{2}}$

$\frac{\sqrt{98 r s^{10}}}{\sqrt{2 r^{3} s^{4}}}$

$\frac{\sqrt{320 m n^{5}}}{\sqrt{45 m^{7} n^{3}}}$

Solution

$\frac{8 n}{3 m^{3}}$

$\frac{\sqrt{810 c^{3} d^{7}}}{\sqrt{1000 c^{5} d}}$

$\frac{\sqrt{98}}{14}$

Solution

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

$\frac{\sqrt{72}}{18}$

$\frac{5 + \sqrt{125}}{15}$

Solution

$\frac{1 + \sqrt{5}}{3}$

$\frac{6 - \sqrt{45}}{12}$

$\frac{\sqrt{96}}{\sqrt{150}}$

Solution

$\frac{4}{5}$

$\frac{\sqrt{28}}{\sqrt{63}}$

$\frac{\sqrt{26 y^{7}}}{\sqrt{2 y}}$

Solution

$y^{3} \sqrt{13}$

$\frac{\sqrt{15 x^{3}}}{\sqrt{3 x}}$

Rationalize a One-Term Denominator

In the following exercises, simplify and rationalize the denominator.

$\frac{10}{\sqrt{6}}$

Solution

$\frac{5 \sqrt{6}}{3}$

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

$\frac{6}{\sqrt{7}}$

Solution

$\frac{6 \sqrt{7}}{7}$

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

$\frac{3}{\sqrt{13}}$

Solution

$\frac{3 \sqrt{13}}{13}$

$\frac{10}{\sqrt{11}}$

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

Solution

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

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

$\frac{4}{9 \sqrt{5}}$

Solution

$\frac{4 \sqrt{5}}{45}$

$\frac{9}{2 \sqrt{7}}$

$- \frac{9}{2 \sqrt{3}}$

Solution

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

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

$\sqrt{\frac{3}{20}}$

Solution

$\frac{\sqrt{15}}{10}$

$\sqrt{\frac{4}{27}}$

$\sqrt{\frac{7}{40}}$

Solution

$\frac{\sqrt{70}}{20}$

$\sqrt{\frac{8}{45}}$

$\sqrt{\frac{19}{175}}$

Solution

$\frac{\sqrt{133}}{35}$

$\sqrt{\frac{17}{192}}$

Rationalize a Two-Term Denominator

In the following exercises, simplify by rationalizing the denominator.

ⓐ $\frac{3}{3 + \sqrt{11}}$ ⓑ $\frac{8}{1 - \sqrt{5}}$

Solution

ⓐ $\frac{3 (3 - \sqrt{11})}{−2}$ ⓑ $−2 (1 + \sqrt{5})$

ⓐ $\frac{4}{4 + \sqrt{7}}$ ⓑ $\frac{7}{2 - \sqrt{6}}$

ⓐ $\frac{5}{5 + \sqrt{6}}$ ⓑ $\frac{6}{3 - \sqrt{7}}$

Solution

ⓐ $\frac{5 (5 - \sqrt{6})}{19}$ ⓑ $3 (3 + \sqrt{7})$

ⓐ $\frac{6}{6 + \sqrt{5}}$ ⓑ $\frac{5}{4 - \sqrt{11}}$

$\frac{\sqrt{3}}{\sqrt{m} - \sqrt{5}}$

Solution

$\frac{\sqrt{3} (\sqrt{m} + \sqrt{5})}{m - 5}$

$\frac{\sqrt{5}}{\sqrt{n} - \sqrt{7}}$

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

Solution

$\frac{\sqrt{2} (\sqrt{x} + \sqrt{6})}{x - 6}$

$\frac{\sqrt{7}}{\sqrt{y} + \sqrt{3}}$

$\frac{\sqrt{r} + \sqrt{5}}{\sqrt{r} - \sqrt{5}}$

Solution

${\frac{(\sqrt{r} + \sqrt{5})}{r - 5}}^{2}$

$\frac{\sqrt{s} - \sqrt{6}}{\sqrt{s} + \sqrt{6}}$

$\frac{\sqrt{150 x^{2} y^{6}}}{\sqrt{6 x^{4} y^{2}}}$

Solution

$\frac{5 y^{2}}{x}$

$\frac{\sqrt{80 p^{3} q}}{\sqrt{5 p q^{5}}}$

$\frac{15}{\sqrt{5}}$

Solution

$3 \sqrt{5}$

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

$\sqrt{\frac{8}{54}}$

Solution

$\frac{2 \sqrt{3}}{9}$

$\sqrt{\frac{12}{20}}$

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

Solution

$\frac{3 (5 - \sqrt{5})}{20}$

$\frac{20}{4 - \sqrt{3}}$

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

Solution

$\frac{\sqrt{2} (\sqrt{x} + \sqrt{3})}{x - 3}$

$\frac{\sqrt{5}}{\sqrt{y} - \sqrt{7}}$

$\frac{\sqrt{x} + \sqrt{8}}{\sqrt{x} - \sqrt{8}}$

Solution

${\frac{(\sqrt{x} + 2 \sqrt{2})}{x - 8}}^{2}$

$\frac{\sqrt{m} - \sqrt{3}}{\sqrt{m} + \sqrt{3}}$

Everyday Math

A supply kit is dropped from an airplane flying at an altitude of 250 feet. Simplify $\sqrt{\frac{250}{16}}$ to determine how many seconds it takes for the supply kit to reach the ground.

Solution

$\frac{5 \sqrt{10}}{4} seconds$

A flare is dropped into the ocean from an airplane flying at an altitude of 1,200 feet. Simplify $\sqrt{\frac{1200}{16}}$ to determine how many seconds it takes for the flare to reach the ocean.

Writing Exercises

ⓐ Simplify $\sqrt{\frac{27}{3}}$ and explain all your steps.
ⓑ Simplify $\sqrt{\frac{27}{5}}$ and explain all your steps.

Solution

Answers will vary.

ⓐ Approximate $\frac{1}{\sqrt{2}}$ by dividing $\frac{1}{1.414}$ using long division without a calculator.
ⓑ Rationalizing the denominator of $\frac{1}{\sqrt{2}}$ gives $\frac{\sqrt{2}}{2} .$ Approximate $\frac{\sqrt{2}}{2}$ by dividing $\frac{1.414}{2}$ using long division without a calculator.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has four columns and four rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no – I don’t get it!” The rows under the column “I can…” read, “divide square roots,” “rationalize a one term denominator.,” and “rationalize a two term denominator.” All the other rows under the columns are empty.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

rationalizing the denominator

The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator.