8.5 Divide Radical Expressions

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 1

Simplify: ⓐ $\frac{\sqrt{72 x^{3}}}{\sqrt{162 x}}$ ⓑ $\frac{\sqrt[3]{32 x^{2}}}{\sqrt[3]{4 x^{5}}} .$

ⓐ

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

ⓑ

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

Simplify: ⓐ $\frac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}$ ⓑ $\frac{\sqrt[3]{−250 m^{} n^{−2}}}{\sqrt[3]{2 m^{−2} n^{4}}} .$

ⓐ

	$\frac{\sqrt{147 a b^{8}}}{\sqrt{3 a^{3} b^{4}}}$
Rewrite using the quotient property.	$\sqrt{\frac{147 a b^{8}}{3 a^{3} b^{4}}}$
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.	$- \frac{5 m}{n^{2}}$

Example 3

Simplify: $\frac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}} .$

	$\frac{\sqrt{54 x^{5} y^{3}}}{\sqrt{3 x^{2} y}}$
Rewrite using the quotient property.	$\sqrt{\frac{54 x^{5} y^{3}}{3 x^{2} y}}$
Remove common factors in the fraction.	$the square root of 18 x cubed y squared$
Rewrite the radicand as a product using the largest perfect square factor.	$\sqrt{9 x^{2} y^{2} \cdot 2 x}$
Rewrite the radical as the product of two radicals.	$\sqrt{9 x^{2} y^{2}} \cdot \sqrt{2 x}$
Simplify.	$3 x y the square root of 2 x$

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 4

Simplify: ⓐ $\frac{4}{\sqrt{3}}$ ⓑ $\sqrt{\frac{3}{20}}$ ⓒ $\frac{3}{\sqrt{6 x}} .$

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.

ⓐ

	$A mathematical expression representing the fraction four divided by the square root of three.$
Multiply both the numerator and denominator by $the square root of 3.$	$A mathematical expression illustrating the process of rationalizing a denominator. The fraction shows (4 * 3) / ( 3 * 3), with the multiplying 3 highlighted in red for clarity.$
Simplify.

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

	$The image displays a mathematical expression: the square root of the fraction 3/20.$
The fraction is not a perfect square, so rewrite using the Quotient Property.	$A mathematical expression displaying a fraction where the numerator is the square root of 3 and the denominator is the square root of 20.$
Simplify the denominator.	$A mathematical expression displaying the fraction: square root of 3 divided by 2 times the square root of 5.$
Multiply the numerator and denominator by $the square root of 5.$
Simplify.
Simplify.	$The mathematical expression shows the square root of 15, divided by 10. It is written as a fraction with 'sqrt(15)' in the numerator and '10' in the denominator.$

ⓒ

	$A mathematical expression showing the fraction 3 over the square root of 6x.$
Multiply the numerator and denominator by $the square root of 6 x.$	$A step in simplifying an algebraic fraction, showing 3 times the square root of 6x divided by the product of the square root of 6x with itself.$
Simplify.	$A mathematical expression showing the fraction 3 times the square root of 6x divided by 6x.$
Simplify.

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,

Two examples of rationalizing denominators are shown. The first example is 1 divided by cube root 2. A note is made that the radicand in the denominator is 1 power of 2 and that we need 2 more to get a perfect cube. We multiply numerator and denominator by the cube root of the quantity 2 squared. The result is cube root 4 divided by cube root of quantity 2 cubed. This simplifies to cube root 4 divided by 2. The second example is 1 divided by fourth root 5. A note is made that the radicand in the denominator is 1 power of 5 and that we need 3 more to get a perfect fourth. We multiply numerator and denominator by the fourth root of the quantity 5 cubed. The result is fourth root of 125 divided by fourth root of quantity 5 to the fourth. This simplifies to fourth root 125 divided by 5.

We will use this technique in the next examples.

Example 5

Simplify ⓐ $\frac{1}{\sqrt[3]{6}}$ ⓑ $\sqrt[3]{\frac{7}{24}}$ ⓒ $\frac{3}{\sqrt[3]{4 x}} .$

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.

ⓐ

	$A mathematical expression showing the fraction 1 over the cube root of 6.$
The radical in the denominator has one factor of 6. Multiply both the numerator and denominator by $the cube root of 6 squared,$ which gives us 2 more factors of 6.	$A mathematical expression displaying a fraction where the numerator is 1 multiplied by the cube root of 6 squared, and the denominator is the cube root of 6 multiplied by the cube root of 6 squared.$
Multiply. Notice the radicand in the denominator has 3 powers of 6.	$A mathematical fraction displaying the cube root of 6 squared over the cube root of 6 cubed.$
Simplify the cube root in the denominator.

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

	$A mathematical expression showing the cube root of the fraction seven over twenty-four.$
The fraction is not a perfect cube, so rewrite using the Quotient Property.	$A fraction with the cube root of 7 in the numerator and the cube root of 24 in the denominator, representing a mathematical expression involving radicals.$
Simplify the denominator.
Multiply the numerator and denominator by $the cube root of 3 squared.$ This will give us 3 factors of 3.	$A fraction with cube root expressions. The numerator is the product of cube root of 7 and cube root of 3 squared. The denominator is 2 times the product of cube root of 3 and cube root of 3 squared.$
Simplify.	$A mathematical fraction with the cube root of 63 over 2 times the cube root of 3 cubed.$
Remember, $\sqrt[3]{3^{3}} = 3 .$
Simplify.

ⓒ

	$A mathematical expression showing the fraction 3 over the cube root of 4x. The numerator is 3, and the denominator is the cube root symbol with 3 as the index, encompassing the term 4x.$
Rewrite the radicand to show the factors.	$A mathematical expression showing the fraction 3 divided by the cube root of (2 squared multiplied by x).$
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.	$A mathematical expression showing a fraction. The numerator is 3 multiplied by the cube root of 2, then multiplied by x squared. The denominator is the cube root of (2 squared times x), multiplied by the cube root of 2, then multiplied by x squared.$
Simplify.	$A mathematical fraction with 3 times the cube root of 2x squared in the numerator, and the cube root of 2 cubed x cubed in the denominator.$
Simplify the radical in the denominator.	$A mathematical expression showing the fraction 3 times the cube root of 2x squared, all over 2x.$

Example 6

Simplify: ⓐ $\frac{1}{\sqrt[4]{2}}$ ⓑ $\sqrt[4]{\frac{5}{64}}$ ⓒ $\frac{2}{\sqrt[4]{8 x}} .$

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.

ⓐ

	$A mathematical expression showing the fraction 1 over the fourth root of 2, often represented as 1 / th root(2). This is equivalent to 2 raised to the power of -1/4.$
The radical in the denominator has one factor of 2. Multiply both the numerator and denominator by $the 4th root of 2 cubed,$ 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.

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

	$A mathematical expression showing the fourth root of the fraction 5 divided by 64.$
The fraction is not a perfect fourth power, so rewrite using the Quotient Property.	$A fraction with the fourth root of 5 in the numerator and the fourth root of 64 in the denominator.$
Rewrite the radicand in the denominator to show the factors.
Simplify the denominator.	$A mathematical expression showing a fraction. The numerator is the fourth root of 5. The denominator is 2 multiplied by the fourth root of 2 squared.$
Multiply the numerator and denominator by $the 4th root of 2 squared.$ This will give us 4 factors of 2.	$A fraction showing an algebraic expression. The numerator is (4th root of 5) * (4th root of 2^2). The denominator is 2 * (4th root of 2^2) * (4th root of 2^2). Some elements are in red.$
Simplify.	$A mathematical fraction is shown. The numerator is the fourth root of 5 multiplied by the fourth root of 4. The denominator is 2 multiplied by the fourth root of 2 to the power of 4.$
Remember, $\sqrt[4]{2^{4}} = 2 .$	$A mathematical expression showing a fraction. The numerator is the fourth root of 20. The denominator is 2 multiplied by 2.$
Simplify.

ⓒ

	$A mathematical expression showing the fraction 2 divided by the fourth root of 8x. The number 2 is in the numerator, and the denominator is the fourth root symbol with 8x inside it.$
Rewrite the radicand to show the factors.	$A mathematical fraction with 2 in the numerator and the fourth root of 2 cubed multiplied by x in the denominator.$
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.	$A mathematical fraction with a numerator of 2 times the fourth root of 2 times x cubed, and a denominator of the fourth root of 2 cubed x times the fourth root of 2 times x cubed.$
Simplify.	$A mathematical expression displaying a fraction. The numerator is 2 multiplied by the fourth root of (2x^3). The denominator is the fourth root of (2^4x^4).$
Simplify the radical in the denominator.	$A mathematical expression displaying a fraction. The numerator is 2 multiplied by the fourth root of 2x cubed. The denominator is 2x.$
Simplify the fraction.

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.

Example 7

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

	$A mathematical expression showing the fraction 5 over 2 minus the square root of 3.$
Multiply the numerator and denominator by the conjugate of the denominator.	$A fraction with 5(2 + sqrt(3)) in the numerator and (2 - sqrt(3))(2 + sqrt(3)) in the denominator, illustrating the process of rationalizing a denominator.$
Multiply the conjugates in the denominator.	$A mathematical fraction with 5(2+sqrt(3)) in the numerator and 2^2 - (sqrt(3))^2 in the denominator.$
Simplify the denominator.	$A mathematical expression showing the fraction 5(2 + square root of 3) divided by 4 - 3.$
Simplify the denominator.
Simplify.

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.

Example 9

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

	$A mathematical expression showing the fraction (square root of x + square root of 7) divided by (square root of x - square root of 7).$
Multiply the numerator and denominator by the conjugate of the denominator.
Multiply the conjugates in the denominator.	$A mathematical expression featuring a fraction. The numerator is (sqrt(x) + sqrt(7)) multiplied by itself, and the denominator is the difference of squares: (sqrt(x))^2 - (sqrt(7))^2.$
Simplify the denominator.	$A mathematical expression showing a fraction. The numerator is (square root of x + square root of 7) squared, and the denominator is x - 7.$

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.

Key Concepts

Quotient Property of Radical Expressions
- If $\sqrt[n]{a}$ and $\sqrt[n]{b}$ are real numbers, $b not equal to 0,$ and for any integer $n greater than or equal to 2$ then,
  $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ and $\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$
Simplified Radical Expressions
- A radical expression is considered simplified if there are:
  - no factors in the radicand that have perfect powers of the index
  - no fractions in the radicand
  - no radicals in the denominator of a fraction

Practice Makes Perfect

Divide Square Roots

In the following exercises, simplify.

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

ⓐ $\frac{4}{3}$ ⓑ $\frac{4}{3}$

ⓐ $\frac{\sqrt{48}}{\sqrt{75}}$ ⓑ $\frac{\sqrt[3]{81}}{\sqrt[3]{24}}$

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

ⓐ $\frac{10 m^{2}}{7}$ ⓑ $\frac{3}{y}$

ⓐ $\frac{\sqrt{108 n^{7}}}{\sqrt{243 n^{3}}}$ ⓑ $\frac{\sqrt[3]{54 y^{}}}{\sqrt[3]{16 y^{4}}}$

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

ⓐ $\frac{5}{6 r^{2}}$ ⓑ $\frac{2 x}{3}$

ⓐ $\frac{\sqrt{196 q^{}}}{\sqrt{484 q^{5}}}$ ⓑ $\frac{\sqrt[3]{16 m^{4}}}{\sqrt[3]{54 m^{}}}$

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

ⓐ $\frac{6 p}{q^{2}}$ ⓑ $- \frac{2 a^{2}}{b}$

ⓐ $\frac{\sqrt{98 r s^{10}}}{\sqrt{2 r^{3} s^{4}}}$ ⓑ $\frac{\sqrt[3]{−375 y^{4} z^{−2}}}{\sqrt[3]{3 y^{−2} z^{4}}}$

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

ⓐ $\frac{8 m^{4}}{3 n^{4}}$ ⓑ $- \frac{2 x^{2}}{3 y^{2}}$

ⓐ $\frac{\sqrt{810 c^{−3} d^{7}}}{\sqrt{1000 c^{} d^{−1}}}$ ⓑ $\frac{\sqrt[3]{24 a^{7} b^{- 1}}}{\sqrt[3]{- 81 a^{- 2} b^{2}}}$

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

$2 x squared the square root of 7 y$

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

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

$2 a b the cube root of 2 a$

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

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

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

ⓐ $\frac{5 \sqrt{6}}{3}$ ⓑ $\frac{2 \sqrt{3}}{9}$ ⓒ $\frac{2 \sqrt{5 x}}{x}$

ⓐ $\frac{8}{\sqrt{3}}$ ⓑ $\sqrt{\frac{7}{40}}$ ⓒ $\frac{8}{\sqrt{2 y}}$

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

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.
ⓒ Why are the two methods of simplifying square roots different?

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.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “divide radical expressions.”, “rationalize a one term denominator”, and “rationalize a two term denominator”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

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

	$A mathematical expression showing the fraction with square root of 3 as the numerator and the difference of square root of u and square root of 6 as the denominator.$
Multiply the numerator and denominator by the conjugate of the denominator.	$A mathematical fraction with a numerator of sqrt(3)(sqrt(u)+sqrt(6)) and a denominator of (sqrt(u)-sqrt(6))(sqrt(u)+sqrt(6)), highlighting the (sqrt(u)+sqrt(6)) term in red.$
Multiply the conjugates in the denominator.	$A fraction with sqrt(3)(sqrt(u)+sqrt(6)) in the numerator and (sqrt(u))^2-(sqrt(6))^2 in the denominator, illustrating a rationalization process.$
Simplify the denominator.	$A mathematical expression showing the fraction sqrt(3)(sqrt(u) + sqrt(6)) over (u - 6).$