# Multiplying and Dividing Radical Expressions

### Learning Outcomes

- Use properties of exponents to multiply radical expressions
- Use properties of exponents to divide radical expressions

**radical expressions**. You can multiply and divide them, too. Multiplying radicals is very simple if the index on all the radicals match. The product raised to a power rule that we discussed previously will help us find products of radical expressions. Recall the rule:

### A Product Raised to a Power Rule

For any numbers*a*and

*b*and any integer

*x*: [latex] {{(ab)}^{x}}={{a}^{x}}\cdot {{b}^{x}}[/latex] For any numbers

*a*and

*b*and any positive integer

*x*: [latex] {{(ab)}^{\frac{1}{x}}}={{a}^{\frac{1}{x}}}\cdot {{b}^{\frac{1}{x}}}[/latex] For any numbers

*a*and

*b*and any positive integer

*x*: [latex] \sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}[/latex]

### Example

Simplify. [latex] \sqrt{18}\cdot \sqrt{16}[/latex]Answer: Use the rule [latex] \sqrt[x]{a}\cdot \sqrt[x]{b}=\sqrt[x]{ab}[/latex] to multiply the radicands.

[latex]\begin{array}{r}\sqrt{18\cdot 16}\\\\\sqrt{288}\end{array}[/latex]

Look for perfect squares in the radicand, and rewrite the radicand as the product of two factors.[latex] \sqrt{144\cdot 2}[/latex]

Identify perfect squares.[latex] \sqrt{{{(12)}^{2}}\cdot 2}[/latex]

Rewrite as the product of two radicals.[latex] \sqrt{{{(12)}^{2}}}\cdot \sqrt{2}[/latex]

Simplify, using [latex] \sqrt{{{x}^{2}}}=\left| x \right|[/latex].[latex]\begin{array}{r}\left| 12 \right|\cdot \sqrt{2}\\12\cdot \sqrt{2}\end{array}[/latex]

#### Answer

[latex-display] \sqrt{18}\cdot \sqrt{16}=12\sqrt{2}[/latex-display]*before*multiplying? In the next example, we will use the same product from above to show that you can simplify before multiplying and get the same result.

### Example

Simplify [latex] \sqrt{18}\cdot \sqrt{16}[/latex]Answer: Look for perfect squares in each radicand, and rewrite as the product of two factors.

[latex] \begin{array}{r}\sqrt{9\cdot 2}\cdot \sqrt{4\cdot 4}\\\\\sqrt{3\cdot 3\cdot 2}\cdot \sqrt{4\cdot 4}\end{array}[/latex]

Identify perfect squares.[latex] \sqrt{{{(3)}^{2}}\cdot 2}\cdot \sqrt{{{(4)}^{2}}}[/latex]

Rewrite as the product of radicals.[latex] \sqrt{{{(3)}^{2}}}\cdot \sqrt{2}\cdot \sqrt{{{(4)}^{2}}}[/latex]

Simplify, using [latex] \sqrt{{{x}^{2}}}=\left| x \right|[/latex].[latex]\begin{array}{c}\left|3\right|\cdot\sqrt{2}\cdot\left|4\right|\\3\cdot\sqrt{2}\cdot4\end{array}[/latex]

Multiply.[latex]12\sqrt{2}[/latex]

#### Answer

[latex-display] \sqrt{18}\cdot \sqrt{16}=12\sqrt{2}[/latex-display]### Try It

[ohm_question]18625[/ohm_question]### Example

Simplify. [latex] \sqrt{12{{x}^{3}}}\cdot \sqrt{3x}[/latex], [latex] x\ge 0[/latex]Answer: Use the rule [latex] \sqrt[x]{a}\cdot \sqrt[x]{b}=\sqrt[x]{ab}[/latex] to multiply the radicands.

[latex] \sqrt{12{{x}^{3}}\cdot 3x}\\\\\sqrt{12\cdot 3\cdot {{x}^{3}}\cdot x}[/latex]

Recall that [latex] {{x}^{3}}\cdot x={{x}^{3+1}}[/latex].[latex]\begin{array}{r}\sqrt{36\cdot {{x}^{3+1}}}\\\\\sqrt{36\cdot {{x}^{4}}}\end{array}[/latex]

Look for perfect squares in the radicand.[latex] \sqrt{{{(6)}^{2}}\cdot {{({{x}^{2}})}^{2}}}[/latex]

Rewrite as the product of the result.[latex] \begin{array}{c}\sqrt{{{(6)}^{2}}}\cdot \sqrt{{{({{x}^{2}})}^{2}}}\\6\cdot {{x}^{2}}\end{array}[/latex]

#### Answer

[latex-display] \sqrt{12{{x}^{3}}}\cdot \sqrt{3x}=6{{x}^{2}}[/latex-display]### Example

Multiply [latex]2\sqrt[3]{18}\cdot-7\sqrt[3]{15}[/latex]Answer: Multiply the factors outside the radicals, and factor the radicands. [latex-display]-14\sqrt[3]{2\cdot3^2}\sqrt[3]{3\cdot5}[/latex-display] Combine the radicands into one radical, and reorganize to see if there are any cubes. [latex-display]-14\sqrt[3]{2\cdot3^2\cdot3\cdot5}=-14\sqrt[3]{2\cdot3^3\cdot5}[/latex-display] Apply the cube root to [latex]3^3[/latex], and simplify the radicand. [latex-display]-14\cdot3\sqrt[3]{2\cdot5}=-42\sqrt[3]{10}[/latex-display]

#### Answer

[latex-display]2\sqrt[3]{18}\cdot-7\sqrt[3]{15}=-42\sqrt[3]{10}[/latex-display]### Example

Multiply [latex]\sqrt[3]{4x^3}\cdot\sqrt[3]{2x^2}[/latex]Answer: Factor the radicands, keeping in mind you want to find cubes. [latex-display]\sqrt[3]{4x^3}\cdot\sqrt[3]{2x^2}=\sqrt[3]{2^2\cdot{x^3}}\cdot\sqrt[3]{2x^2}[/latex-display] Combine the radicands into one radical, and reorganize into cubes where possible. [latex-display]\begin{array}{c}\sqrt[3]{2^2\cdot{x^3}}\cdot\sqrt[3]{2x^2}\\\\=\sqrt[3]{2^2\cdot{x^3}\cdot2\cdot{x^2}}\\\\=\sqrt[3]{2^3\cdot{x^3}\cdot{x^2}}\end{array}[/latex-display] Apply the cube root to [latex]2^3[/latex], and [latex]x^3[/latex] and simplify the radicand. [latex-display]\sqrt[3]{2^3\cdot{x^3}\cdot{x^2}}=2\cdot{x}\sqrt[3]{x^2}[/latex-display]

#### Answer

[latex-display]\sqrt[3]{4x^3}\cdot\sqrt[3]{2x^2}=2\cdot{x}\sqrt[3]{x^2}[/latex-display]### Example

Simplify. [latex] \sqrt[3]{{{x}^{5}}{{y}^{2}}}\cdot 5\sqrt[3]{8{{x}^{2}}{{y}^{4}}}[/latex]Answer:
Notice that *both* radicals are cube roots, so you can use the rule [latex] [/latex] to multiply the radicands.

[latex]\begin{array}{l}5\sqrt[3]{{{x}^{5}}{{y}^{2}}\cdot 8{{x}^{2}}{{y}^{4}}}\\5\sqrt[3]{8\cdot {{x}^{5}}\cdot {{x}^{2}}\cdot {{y}^{2}}\cdot {{y}^{4}}}\\5\sqrt[3]{8\cdot {{x}^{5+2}}\cdot {{y}^{2+4}}}\\5\sqrt[3]{8\cdot {{x}^{7}}\cdot {{y}^{6}}}\end{array}[/latex]

Look for perfect cubes in the radicand. Since [latex] {{x}^{7}}[/latex] is not a perfect cube, it has to be rewritten as [latex] {{x}^{6+1}}={{({{x}^{2}})}^{3}}\cdot x[/latex].[latex] 5\sqrt[3]{{{(2)}^{3}}\cdot {{({{x}^{2}})}^{3}}\cdot x\cdot {{({{y}^{2}})}^{3}}}[/latex]

Rewrite as the product of radicals.[latex] \begin{array}{r}5\sqrt[3]{{{(2)}^{3}}}\cdot \sqrt[3]{{{({{x}^{2}})}^{3}}}\cdot \sqrt[3]{{{({{y}^{2}})}^{3}}}\cdot \sqrt[3]{x}\\5\cdot 2\cdot {{x}^{2}}\cdot {{y}^{2}}\cdot \sqrt[3]{x}\end{array}[/latex]

The answer is [latex]10{{x}^{2}}{{y}^{2}}\sqrt[3]{x}[/latex].### Example

Simplify. [latex] 2\sqrt[4]{16{{x}^{9}}}\cdot \sqrt[4]{{{y}^{3}}}\cdot \sqrt[4]{81{{x}^{3}}y}[/latex], [latex] x\ge 0[/latex], [latex] y\ge 0[/latex]Answer: Notice this expression is multiplying three radicals with the same (fourth) root. Simplify each radical, if possible, before multiplying. Be looking for powers of [latex]4[/latex] in each radicand.

[latex] 2\sqrt[4]{{{(2)}^{4}}\cdot {{({{x}^{2}})}^{4}}\cdot x}\cdot \sqrt[4]{{{y}^{3}}}\cdot \sqrt[4]{{{(3)}^{4}}\cdot {{x}^{3}}y}[/latex]

Rewrite as the product of radicals.[latex] 2\sqrt[4]{{{(2)}^{4}}}\cdot \sqrt[4]{{{({{x}^{2}})}^{4}}}\cdot \sqrt[4]{x}\cdot \sqrt[4]{{{y}^{3}}}\cdot \sqrt[4]{{{(3)}^{4}}}\cdot \sqrt[4]{{{x}^{3}}y}[/latex]

Identify and pull out powers of [latex]4[/latex], using the fact that [latex] \sqrt[4]{{{x}^{4}}}=\left| x \right|[/latex].[latex] \begin{array}{r}2\cdot \left| 2 \right|\cdot \left| {{x}^{2}} \right|\cdot \sqrt[4]{x}\cdot \sqrt[4]{{{y}^{3}}}\cdot \left| 3 \right|\cdot \sqrt[4]{{{x}^{3}}y}\\2\cdot 2\cdot {{x}^{2}}\cdot \sqrt[4]{x}\cdot \sqrt[4]{{{y}^{3}}}\cdot 3\cdot \sqrt[4]{{{x}^{3}}y}\end{array}[/latex]

Since all the radicals are fourth roots, you can use the rule [latex] \sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}[/latex] to multiply the radicands.[latex]\begin{array}{r}2\cdot 2\cdot 3\cdot {{x}^{2}}\cdot \sqrt[4]{x\cdot {{y}^{3}}\cdot {{x}^{3}}y}\\12{{x}^{2}}\sqrt[4]{{{x}^{1+3}}\cdot {{y}^{3+1}}}\end{array}[/latex]

Now that the radicands have been multiplied, look again for powers of [latex]4[/latex], and pull them out. We can drop the absolute value signs in our final answer because at the start of the problem we were told [latex] x\ge 0[/latex], [latex] y\ge 0[/latex].[latex] \begin{array}{l}12{{x}^{2}}\sqrt[4]{{{x}^{4}}\cdot {{y}^{4}}}\\12{{x}^{2}}\sqrt[4]{{{x}^{4}}}\cdot \sqrt[4]{{{y}^{4}}}\\12{{x}^{2}}\cdot \left| x \right|\cdot \left| y \right|\end{array}[/latex]

The answer is [latex]12{{x}^{3}}y,\,\,x\ge 0,\,\,y\ge 0[/latex].## Dividing Radical Expressions

You can use the same ideas to help you figure out how to simplify and divide radical expressions. Recall that the Product Raised to a Power Rule states that [latex] \sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}[/latex]. Well, what if you are dealing with a quotient instead of a product? There is a rule for that, too. The**Quotient Raised to a Power Rule**states that [latex] {{\left( \frac{a}{b} \right)}^{x}}=\frac{{{a}^{x}}}{{{b}^{x}}}[/latex]. Again, if you imagine that the exponent is a rational number, then you can make this rule applicable for roots as well:

[latex] {{\left( \frac{a}{b} \right)}^{\frac{1}{x}}}=\frac{{{a}^{\frac{1}{x}}}}{{{b}^{\frac{1}{x}}}}[/latex]

Therefore

[latex] \sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}[/latex].

### A Quotient Raised to a Power Rule

For any real numbers*a*and

*b*(

*b*≠ 0) and any positive integer

*x*: [latex] {{\left( \frac{a}{b} \right)}^{\frac{1}{x}}}=\frac{{{a}^{\frac{1}{x}}}}{{{b}^{\frac{1}{x}}}}[/latex] For any real numbers

*a*and

*b*(

*b*≠ 0) and any positive integer

*x*: [latex] \sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}[/latex]

### Example

Simplify. [latex] \sqrt{\frac{48}{25}}[/latex]Answer: Use the rule [latex] \sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}[/latex] to create two radicals; one in the numerator and one in the denominator.

[latex] \frac{\sqrt{48}}{\sqrt{25}}[/latex]

Simplify each radical. Look for perfect square factors in the radicand, and rewrite the radicand as a product of factors.[latex] \begin{array}{c}\frac{\sqrt{16\cdot 3}}{\sqrt{25}}\\\\\text{or}\\\\\frac{\sqrt{4\cdot 4\cdot 3}}{\sqrt{5\cdot 5}}\end{array}[/latex]

Identify and pull out perfect squares.[latex] \begin{array}{r}\frac{\sqrt{{{(4)}^{2}}\cdot 3}}{\sqrt{{{(5)}^{2}}}}\\\\\frac{\sqrt{{{(4)}^{2}}}\cdot \sqrt{3}}{\sqrt{{{(5)}^{2}}}}\end{array}[/latex]

Simplify.[latex] \frac{4\cdot \sqrt{3}}{5}[/latex]

#### Answer

[latex-display] \sqrt{\frac{48}{25}}=\frac{4\sqrt{3}}{5}[/latex-display]### Example

Simplify. [latex] \sqrt[3]{\frac{640}{40}}[/latex]Answer: Rewrite using the Quotient Raised to a Power Rule.

[latex] \frac{\sqrt[3]{640}}{\sqrt[3]{40}}[/latex]

Simplify each radical. Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors.[latex] \frac{\sqrt[3]{64\cdot 10}}{\sqrt[3]{8\cdot 5}}[/latex]

Identify and pull out perfect cubes.[latex] \begin{array}{r}\frac{\sqrt[3]{{{(4)}^{3}}\cdot 10}}{\sqrt[3]{{{(2)}^{3}}\cdot 5}}\\\\\frac{\sqrt[3]{{{(4)}^{3}}}\cdot \sqrt[3]{10}}{\sqrt[3]{{{(2)}^{3}}}\cdot \sqrt[3]{5}}\\\\\frac{4\cdot \sqrt[3]{10}}{2\cdot \sqrt[3]{5}}\end{array}[/latex]

You can simplify this expression even further by looking for common factors in the numerator and denominator.[latex] \frac{4\sqrt[3]{10}}{2\sqrt[3]{5}}[/latex]

Rewrite the numerator as a product of factors.[latex] \frac{2\cdot 2\sqrt[3]{5}\cdot \sqrt[3]{2}}{2\sqrt[3]{5}}[/latex]

Identify factors of [latex]1[/latex], and simplify.[latex] \begin{array}{r}2\cdot \frac{2\sqrt[3]{5}}{2\sqrt[3]{5}}\cdot \sqrt[3]{2}\\\\2\cdot 1\cdot \sqrt[3]{2}\end{array}[/latex]

The answer is [latex]2\sqrt[3]{2}[/latex].### Example

Simplify. [latex] \frac{\sqrt[3]{640}}{\sqrt[3]{40}}[/latex]Answer: Since both radicals are cube roots, you can use the rule [latex] \frac{\sqrt[x]{a}}{\sqrt[x]{b}}=\sqrt[x]{\frac{a}{b}}[/latex] to create a single rational expression underneath the radical.

[latex] \sqrt[3]{\frac{640}{40}}[/latex]

Within the radical, divide [latex]640[/latex] by [latex]40[/latex].[latex] \begin{array}{r}640\div 40=16\\\sqrt[3]{16}\end{array}[/latex]

Look for perfect cubes in the radicand, and rewrite the radicand as a product of factors.[latex]\sqrt[3]{8\cdot2}[/latex]

Identify perfect cubes and pull them out.[latex] \begin{array}{r}\sqrt[3]{{{(2)}^{3}}\cdot 2}\\\sqrt[3]{{(2)}^{3}}\cdot\sqrt[3]{2}\end{array}[/latex]

Simplify.[latex]2\cdot\sqrt[3]{2}[/latex]

The answer is [latex]2\sqrt[3]{2}[/latex].### Example

Simplify. [latex]\frac{\sqrt{30x}}{\sqrt{10x}},x>0[/latex]Answer: Use the Quotient Raised to a Power Rule to rewrite this expression.

[latex]\sqrt{\frac{30x}{10x}}[/latex]

Simplify [latex] \sqrt{\frac{30x}{10x}}[/latex] by identifying similar factors in the numerator and denominator and then identifying factors of [latex]1[/latex].[latex]\begin{array}{r}\sqrt{\frac{3\cdot10x}{10x}}\\\\\sqrt{3\cdot\frac{10x}{10x}}\\\\\sqrt{3\cdot1}\end{array}[/latex]

#### Answer

[latex-display] \frac{\sqrt{30x}}{\sqrt{10x}}=\sqrt{3}[/latex-display]As you become more familiar with dividing and simplifying radical expressions, make sure you continue to pay attention to the roots of the radicals that you are dividing. For example, you can think of this expression:

[latex] \frac{\sqrt{8{{y}^{2}}}}{\sqrt{225{{y}^{4}}}}[/latex]

As equivalent to:

[latex] \sqrt{\frac{8{{y}^{2}}}{225{{y}^{4}}}}[/latex]

This is because both the numerator and the denominator are square roots.

Notice that you cannot express this expression:

[latex] \frac{\sqrt{8{{y}^{2}}}}{\sqrt[4]{225{{y}^{4}}}}[/latex]

In this format:

[latex] \sqrt[4]{\frac{8{{y}^{2}}}{225{{y}^{4}}}}[/latex].

This is because the numerator is a square root and the denominator is a fourth root. In this last video, we show more examples of simplifying a quotient with radicals.### Example

Simplify. [latex] \frac{\sqrt[3]{24x{{y}^{4}}}}{\sqrt[3]{8y}},\,\,y\ne 0[/latex]Answer: Use the Quotient Raised to a Power Rule to rewrite this expression.

[latex] \sqrt[3]{\frac{24x{{y}^{4}}}{8y}}[/latex]

Simplify [latex] \sqrt[3]{\frac{24x{{y}^{4}}}{8y}}[/latex] by identifying similar factors in the numerator and denominator and then identifying factors of [latex]1[/latex].[latex]\begin{array}{l}\sqrt[3]{\frac{8\cdot 3\cdot x\cdot {{y}^{3}}\cdot y}{8\cdot y}}\\\\\sqrt[3]{\frac{3\cdot x\cdot {{y}^{3}}}{1}\cdot \frac{8y}{8y}}\\\\\sqrt[3]{\frac{3\cdot x\cdot {{y}^{3}}}{1}\cdot 1}\end{array}[/latex]

Identify perfect cubes and pull them out of the radical.[latex] \sqrt[3]{3x{{y}^{3}}}\\\sqrt[3]{{{(y)}^{3}}\cdot \,3x}[/latex]

Simplify.[latex] \sqrt[3]{{{(y)}^{3}}}\cdot \,\sqrt[3]{3x}[/latex]

The answer is [latex]y\,\sqrt[3]{3x}[/latex].### Try It

[ohm_question]91537[/ohm_question]## Summary

The Product Raised to a Power Rule and the Quotient Raised to a Power Rule can be used to simplify radical expressions as long as the roots of the radicals are the same. The Product Rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. The same is true of roots: [latex] \sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}[/latex]. When dividing radical expressions, the rules governing quotients are similar: [latex] \sqrt[x]{\frac{a}{b}}=\frac{\sqrt[x]{a}}{\sqrt[x]{b}}[/latex].## Contribute!

## Licenses & Attributions

### CC licensed content, Original

- Screenshot: Multiply and divide.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Revision and Adaptation.
**Provided by:**Lumen Learning**License:**CC BY: Attribution. - Multiply Square Roots.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution. - Multiple Cube Roots.
**Authored by:**James Sousa (Mathispower4u.com) for Lumen Learning.**License:**CC BY: Attribution.

### CC licensed content, Shared previously

- Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program.
**Provided by:**Monterey Institute of Technology and Education**Located at:**https://www.nroc.org/.**License:**CC BY: Attribution.