Divide Radical Expressions

We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals.

We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars re needed.

We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.

Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator 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. 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.

This process is still used today, and is useful in other areas of mathematics, too.

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 radical.

Similarly, a radical expression is not considered simplified if the radicand contains a fraction.

To rationalize a denominator with a square root, we use the property that

{(\sqrt{a})}^{2} = a .

Simplify: ⓐ $\frac{4}{\sqrt{3}}$

ⓑ $\sqrt{\frac{3}{20}}$

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

ⓐ* * *


{: valign=”top”}	Multiply both the numerator and denominator by $\sqrt{3} .$

| | {: valign=”top”}| Simplify. | | {: valign=”top”}{: .unnumbered .unstyled .can-break summary=”To rationalize the denominator of 4 divided by square root 3 we multiply both the numerator and denominator by square root 3. The result is the 4 times square root 3 divided by the quantity square root 3 times square root 3 in parentheses. Simplifying we get 4 times square root 3 divided by 3.” data-label=””}

ⓑ We always simplify the radical in the denominator first, before we rationalize it. This way the numbers stay smaller and easier to work with.


The fraction is not a perfect square, so rewrite using the Quotient Property.
Simplify the denominator.
Multiply the numerator and denominator by $\sqrt{5} .$
Simplify.
Simplify.

ⓒ* * *


{: valign=”top”}	Multiply the numerator and denominator by $\sqrt{6 x} .$

| | {: valign=”top”}| Simplify. | | {: valign=”top”}| Simplify. | | {: valign=”top”}{: .unnumbered .unstyled .can-break summary=”To rationalize the denominator of 3 divided by square root of the quantity 6 x in parentheses we multiply both the numerator and denominator by square root of the quantity 6 x in parentheses. This is written out as 3 times square root of the quantity 6 x in parentheses divided by the quantity square root 6 x times square root 6 x in parentheses. The result is 3 times square root of the quantity 6 x in parentheses divided by the quantity 6 x in parentheses. Simplifying we square root of the quantity 6 x in parentheses divided by the quantity 2 x.” data-label=””}

When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.

We will follow a similar process to rationalize higher roots. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index. When we simplify the new radical, the denominator will no longer have a radical.

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.

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

Notice we did not distribute the 5 in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.

Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.

Key Concepts

Practice Makes Perfect

Divide Square Roots

In the following exercises, simplify.

ⓐ $\frac{\sqrt{128}}{\sqrt{72}}$

ⓑ $\frac{\sqrt[3]{128}}{\sqrt[3]{54}}$

ⓐ $\frac{4}{3}$

ⓑ $\frac{4}{3}$

ⓐ $\frac{\sqrt{48}}{\sqrt{75}}$

ⓑ $\frac{\sqrt[3]{81}}{\sqrt[3]{24}}$

ⓐ $\frac{\sqrt{200 m^{5}}}{\sqrt{98 m}}$

ⓑ $\frac{\sqrt[3]{54 y^{2}}}{\sqrt[3]{2 y^{5}}}$

ⓐ $\frac{10 m^{2}}{7}$

ⓑ $\frac{3}{y}$

ⓐ $\frac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}$

ⓑ $\frac{\sqrt[3]{54 y^{}}}{\sqrt[3]{16 y^{4}}}$

ⓐ $\frac{\sqrt{75 r^{3}}}{\sqrt{108 r^{7}}}$

ⓑ $\frac{\sqrt[3]{24 x^{7}}}{\sqrt[3]{81 x^{4}}}$

ⓐ $\frac{5}{6 r^{2}}$

ⓑ $\frac{2 x}{3}$

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

ⓑ $\frac{\sqrt[3]{16 m^{4}}}{\sqrt[3]{54 m^{}}}$

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

ⓑ $\frac{\sqrt[3]{−16 a^{4} b^{−2}}}{\sqrt[3]{2 a^{−2} b^{}}}$

ⓐ $\frac{6 p}{q^{2}}$

ⓑ $- \frac{2 a^{2}}{b}$

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

ⓑ $\frac{\sqrt[3]{−375 y^{4} z^{−2}}}{\sqrt[3]{3 y^{−2} z^{4}}}$

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

ⓑ $\frac{\sqrt[3]{16 x^{4} y^{−2}}}{\sqrt[3]{−54 x^{−2} y^{4}}}$

ⓐ $\frac{8 m^{4}}{3 n^{4}}$

ⓑ $- \frac{x^{2}}{2 y^{2}}$

ⓐ $\frac{\sqrt{810 c^{−3} d^{7}}}{\sqrt{1000 c^{} d^{−1}}}$

ⓑ $\frac{\sqrt[3]{24 a^{7} b^{- 1}}}{\sqrt[3]{- 81 a^{- 2} b^{2}}}$

\frac{\sqrt{56 x^{5} y^{4}}}{\sqrt{2 x y^{3}}}

4 x^{4} \sqrt{7 y}

\frac{\sqrt{72 a^{3} b^{6}}}{\sqrt{3 a b^{3}}}

\frac{\sqrt[3]{48 a^{3} b^{6}}}{\sqrt[3]{3 a^{- 1} b^{3}}}

2 a b \sqrt[3]{2 a}

\frac{\sqrt[3]{162 x^{- 3} y^{6}}}{\sqrt[3]{2 x^{3} y^{- 2}}}

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

ⓐ $\frac{10}{\sqrt{6}}$

ⓑ $\sqrt{\frac{4}{27}}$

ⓐ $\frac{5 \sqrt{6}}{3}$

ⓑ $\frac{2 \sqrt{3}}{9}$

ⓐ $\frac{8}{\sqrt{3}}$

ⓑ $\sqrt{\frac{7}{40}}$

ⓐ $\frac{6}{\sqrt{7}}$

ⓑ $\sqrt{\frac{8}{45}}$

ⓐ $\frac{6 \sqrt{7}}{7}$

ⓑ $\frac{2 \sqrt{10}}{15}$

ⓐ $\frac{4}{\sqrt{5}}$

ⓑ $\sqrt{\frac{27}{80}}$

ⓐ $\frac{1}{\sqrt[3]{5}}$

ⓑ $\sqrt[3]{\frac{5}{24}}$

ⓐ $\frac{\sqrt[3]{25}}{5}$

ⓑ $\frac{\sqrt[3]{45}}{6}$

ⓐ $\frac{1}{\sqrt[3]{3}}$

ⓑ $\sqrt[3]{\frac{5}{32}}$

ⓐ $\frac{1}{\sqrt[3]{11}}$

ⓑ $\sqrt[3]{\frac{7}{54}}$

ⓐ $\frac{\sqrt[3]{121}}{11}$

ⓑ $\frac{\sqrt[3]{28}}{6}$

ⓐ $\frac{1}{\sqrt[3]{13}}$

ⓑ $\sqrt[3]{\frac{3}{128}}$

ⓐ $\frac{1}{\sqrt[4]{7}}$

ⓑ $\sqrt[4]{\frac{5}{32}}$

ⓐ $\frac{\sqrt[4]{343}}{7}$

ⓑ $\frac{\sqrt[4]{40}}{4}$

ⓐ $\frac{1}{\sqrt[4]{4}}$

ⓑ $\sqrt[4]{\frac{9}{32}}$

ⓐ $\frac{1}{\sqrt[4]{9}}$

ⓑ $\sqrt[4]{\frac{25}{128}}$

ⓐ $\frac{\sqrt[4]{9}}{3}$

ⓑ $\frac{\sqrt[4]{50}}{4}$

ⓐ $\frac{1}{\sqrt[4]{8}}$

ⓑ $\sqrt[4]{\frac{27}{128}}$

Rationalize a Two Term Denominator

In the following exercises, simplify.

\frac{8}{1 - \sqrt{5}}

−2 (1 + \sqrt{5})

\frac{7}{2 - \sqrt{6}}

\frac{6}{3 - \sqrt{7}}

3 (3 + \sqrt{7})

\frac{5}{4 - \sqrt{11}}

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

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

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

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

\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}}

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

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

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

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

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

Writing Exercises

ⓐ Simplify $\sqrt{\frac{27}{3}}$

and explain all your steps.* * *

ⓑ Simplify $\sqrt{\frac{27}{5}}$

and explain all your steps.* * *

Answers will vary.

Explain what is meant by the word rationalize in the phrase, “rationalize a denominator.”

Explain why multiplying $\sqrt{2 x} - 3$

by its conjugate results in an expression with no radicals.

Answers will vary.

Explain why multiplying $\frac{7}{\sqrt[3]{x}}$

by $\frac{\sqrt[3]{x}}{\sqrt[3]{x}}$

does not rationalize the denominator.

Self Check

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

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


Rewrite the radicand to show the factors.
Multiply the numerator and denominator by $\sqrt[3]{2 \cdot x^{2}} .$ This will get us 3 factors of 2 and 3 factors of x.
Simplify.
Simplify the radical in the denominator.


The fraction is not a perfect fourth power, so rewrite using the Quotient Property.
Rewrite the radicand in the denominator to show the factors.
Simplify the denominator.
Multiply the numerator and denominator by $\sqrt[4]{2^{2}} .$ This will give us 4 factors of 2.
Simplify.
Remember, $\sqrt[4]{2^{4}} = 2 .$
Simplify.


Rewrite the radicand to show the factors.
Multiply the numerator and denominator by $\sqrt[4]{2 \cdot x^{3}} .$ This will get us 4 factors of 2 and 4 factors of x.
Simplify.
Simplify the radical in the denominator.
Simplify the fraction.


The radical in the denominator has one factor of 6. Multiply both the numerator and denominator by $\sqrt[3]{6^{2}},$ which gives us 2 more factors of 6.
Multiply. Notice the radicand in the denominator has 3 powers of 6.
Simplify the cube root in the denominator.


The fraction is not a perfect cube, so rewrite using the Quotient Property.
Simplify the denominator.
Multiply the numerator and denominator by $\sqrt[3]{3^{2}} .$ This will give us 3 factors of 3.
Simplify.
Remember, $\sqrt[3]{3^{3}} = 3 .$
Simplify.


The radical in the denominator has one factor of 2. Multiply both the numerator and denominator by $\sqrt[4]{2^{3}},$ which gives us 3 more factors of 2.
Multiply. Notice the radicand in the denominator has 4 powers of 2.
Simplify the fourth root in the denominator.


Multiply the numerator and denominator by the conjugate of the denominator.
Multiply the conjugates in the denominator.
Simplify the denominator.
Simplify the denominator.
Simplify.