8.5 Divide Radical Expressions

Learning objectives

Learning objectives

• Divide radical expressions

• Rationalize a one term denominator

• Rationalize a two term denominator

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 are 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.

Example 8.47

Solution

ⓐ

$\,\frac{\sqrt{72{x}^{3}}}{\sqrt{162x}}$

Rewrite using the quotient property,$\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}.$ · $\,\sqrt{\frac{72{x}^{3}}{162x}}$ 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. · $\,\frac{2x}{3}$

ⓑ

$\,\frac{\sqrt[3]{32{x}^{2}}}{\sqrt[3]{4{x}^{5}}}$

Rewrite using the quotient property,$\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}.$ · $\,\sqrt[3]{\frac{32{x}^{2}}{4{x}^{5}}}$ Simplify the fraction under the radical. · $\,\sqrt[3]{\frac{8}{{x}^{3}}}$ Simplify the radical. · $\,\frac{2}{x}$

Example 8.48

Solution

ⓐ

$\,\frac{\sqrt{147a{b}^{8}}}{\sqrt{3{a}^{3}{b}^{4}}}$

Rewrite using the quotient property. · $\,\sqrt{\frac{147a{b}^{8}}{3{a}^{3}{b}^{4}}}$ Remove common factors in the fraction. · $\,\sqrt{\frac{49{b}^{4}}{{a}^{2}}}$ Simplify the radical. · $\,\frac{7{b}^{2}}{a}$

ⓑ

$\,\frac{\sqrt[3]{-250{m}^{}{n}^{-2}}}{\sqrt[3]{2{m}^{-2}{n}^{4}}}$

Rewrite using the quotient property. · $\,\sqrt[3]{\frac{-250{m}^{}{n}^{-2}}{2{m}^{-2}{n}^{4}}}$ Simplify the fraction under the radical. · $\,\sqrt[3]{\frac{-125{m}^{3}}{{n}^{6}}}$ Simplify the radical. · $\,-\frac{5m}{{n}^{2}}$

Example 8.49

Solution

$\,\frac{\sqrt{54{x}^{5}{y}^{3}}}{\sqrt{3{x}^{2}y}}$

Rewrite using the quotient property. · $\,\sqrt{\frac{54{x}^{5}{y}^{3}}{3{x}^{2}y}}$ Remove common factors in the fraction. · $\,\sqrt{18{x}^{3}{y}^{2}}$ Rewrite the radicand as a productusing the largest perfect square factor. · $\,\sqrt{9{x}^{2}{y}^{2}\cdot{}2x}$ Rewrite the radical as the product of tworadicals. · $\,\sqrt{9{x}^{2}{y}^{2}}\cdot\sqrt{2x}$ Simplify. · $\,3xy\sqrt{2x}$

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

Example 8.50

Solution

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.

ⓐ

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

ⓒ

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.

For example,

We will use this technique in the next examples.

Example 8.51

Solution

To rationalize a denominator with a cube root, we can multiply by a cube root that will give us a perfect cube in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

ⓐ

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

ⓒ

Example 8.52

Solution

To rationalize a denominator with a fourth root, we can multiply by a fourth root that will give us a perfect fourth power in the radicand in the denominator. To keep the fraction equivalent, we multiply both the numerator and denominator by the same factor.

ⓐ

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

ⓒ

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.

Example 8.53

Solution

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.

Example 8.54

Solution

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

Example 8.55

Solution

We do not square the numerator. Leaving it in factored form, we can see there are no common factors to remove from the numerator and denominator.

Section 8.5 Exercises

Section 8.5 Exercises

Divide Square Roots

In the following exercises, simplify.

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Rationalize a Two Term Denominator

In the following exercises, simplify.

Writing Exercises

ⓐ 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?

1. Simplify: ⓐ $\frac{\sqrt{50{s}^{3}}}{\sqrt{128s}}$ ⓑ $\frac{\sqrt[3]{56{a}^{}}}{\sqrt[3]{7{a}^{4}}}.$

2. Simplify: ⓐ $\frac{\sqrt{75{q}^{5}}}{\sqrt{108q}}$ ⓑ $\frac{\sqrt[3]{72{b}^{2}}}{\sqrt[3]{9{b}^{5}}}.$

3. Simplify: ⓐ $\frac{\sqrt{162{x}^{10}{y}^{2}}}{\sqrt{2{x}^{6}{y}^{6}}}$ ⓑ $\frac{\sqrt[3]{-128{x}^{2}{y}^{-1}}}{\sqrt[3]{2{x}^{-1}{y}^{2}}}.$

1. Simplify: ⓐ $\frac{\sqrt{300{m}^{3}{n}^{7}}}{\sqrt{3{m}^{5}n}}$ ⓑ $\frac{\sqrt[3]{-81p{q}^{-1}}}{\sqrt[3]{3{p}^{-2}{q}^{5}}}.$

2. Simplify: $\frac{\sqrt{64{x}^{4}{y}^{5}}}{\sqrt{2x{y}^{3}}}.$

3. Simplify: $\frac{\sqrt{96{a}^{5}{b}^{4}}}{\sqrt{2{a}^{3}b}}.$

1. Simplify: ⓐ $\frac{5}{\sqrt{3}}$ ⓑ $\sqrt{\frac{3}{32}}$ ⓒ $\frac{2}{\sqrt{2x}}.$

2. Simplify: ⓐ $\frac{6}{\sqrt{5}}$ ⓑ $\sqrt{\frac{7}{18}}$ ⓒ $\frac{5}{\sqrt{5x}}.$

3. Simplify: ⓐ $\frac{1}{\sqrt[3]{7}}$ ⓑ $\sqrt[3]{\frac{5}{12}}$ ⓒ $\frac{5}{\sqrt[3]{9y}}.$

1. Simplify: ⓐ $\frac{1}{\sqrt[3]{2}}$ ⓑ $\sqrt[3]{\frac{3}{20}}$ ⓒ $\frac{2}{\sqrt[3]{25n}}.$

2. Simplify: ⓐ $\frac{1}{\sqrt[4]{3}}$ ⓑ $\sqrt[4]{\frac{3}{64}}$ ⓒ $\frac{3}{\sqrt[4]{125x}}.$

3. Simplify: ⓐ $\frac{1}{\sqrt[4]{5}}$ ⓑ $\sqrt[4]{\frac{7}{128}}$ ⓒ $\frac{4}{\sqrt[4]{4x}}$

1. Simplify: $\frac{3}{1-\sqrt{5}}.$

2. Simplify: $\frac{2}{4-\sqrt{6}}.$

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

1. Simplify: $\frac{\sqrt{10}}{\sqrt{y}-\sqrt{3}}.$

2. Simplify: $\frac{\sqrt{p}+\sqrt{2}}{\sqrt{p}-\sqrt{2}}.$

3. Simplify: $\frac{\sqrt{q}-\sqrt{10}}{\sqrt{q}+\sqrt{10}}$

1. ⓐ $\frac{\sqrt{128}}{\sqrt{72}}$ ⓑ $\frac{\sqrt[3]{128}}{\sqrt[3]{54}}$

2. ⓐ $\frac{\sqrt{200{m}^{5}}}{\sqrt{98m}}$ ⓑ $\frac{\sqrt[3]{54{y}^{2}}}{\sqrt[3]{2{y}^{5}}}$

3. ⓐ $\frac{\sqrt{75{r}^{3}}}{\sqrt{108{r}^{7}}}$ ⓑ $\frac{\sqrt[3]{24{x}^{7}}}{\sqrt[3]{81{x}^{4}}}$

1. ⓐ $\frac{\sqrt{108{p}^{5}{q}^{2}}}{\sqrt{3{p}^{3}{q}^{6}}}$ ⓑ $\frac{\sqrt[3]{-16{a}^{4}{b}^{-2}}}{\sqrt[3]{2{a}^{-2}{b}^{}}}$

2. ⓐ $\frac{\sqrt{320m{n}^{-5}}}{\sqrt{45{m}^{-7}{n}^{3}}}$ ⓑ $\frac{\sqrt[3]{16{x}^{4}{y}^{-2}}}{\sqrt[3]{-54{x}^{-2}{y}^{4}}}$

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

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

2. ⓐ $\frac{10}{\sqrt{6}}$ ⓑ $\sqrt{\frac{4}{27}}$ ⓒ $\frac{10}{\sqrt{5x}}$

3. ⓐ $\frac{6}{\sqrt{7}}$ ⓑ $\sqrt{\frac{8}{45}}$ ⓒ $\frac{12}{\sqrt{3p}}$

1. ⓐ $\frac{1}{\sqrt[3]{5}}$ ⓑ $\sqrt[3]{\frac{5}{24}}$ ⓒ $\frac{4}{\sqrt[3]{36a}}$

2. ⓐ $\frac{1}{\sqrt[3]{11}}$ ⓑ $\sqrt[3]{\frac{7}{54}}$ ⓒ $\frac{3}{\sqrt[3]{3{x}^{2}}}$

3. ⓐ $\frac{1}{\sqrt[4]{7}}$ ⓑ $\sqrt[4]{\frac{5}{32}}$ ⓒ $\frac{4}{\sqrt[4]{4{x}^{2}}}$

1. ⓐ $\frac{1}{\sqrt[4]{9}}$ ⓑ $\sqrt[4]{\frac{25}{128}}$ ⓒ $\frac{6}{\sqrt[4]{27a}}$

2. $\frac{8}{1-\sqrt{5}}$

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

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

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

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

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

2. ⓐ Simplify $\sqrt{\frac{27}{3}}$ and explain all your steps.ⓑ Simplify $\sqrt{\frac{27}{5}}$ and explain all your steps.ⓒ Why are the two methods of simplifying square roots different?

3. Explain why multiplying $\sqrt{2x}-3$ by its conjugate results in an expression with no radicals.