Multiply and Divide Radical Expressions
Learning Outcomes
- Multiply and divide radical expressions
- Use the product raised to a power rule to multiply radical expressions
- Use the quotient raised to a power rule to divide radical expressions
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]
The answer is [latex]12\sqrt{2}[/latex].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]
Example
Simplify. [latex] \sqrt{12{{x}^{4}}}\cdot \sqrt{3x^2}[/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}^{4}}\cdot 3x^2}\\\sqrt{12\cdot 3\cdot {{x}^{4}}\cdot x^2}[/latex]
Recall that [latex] {{x}^{4}}\cdot x^2={{x}^{4+2}}[/latex].[latex]\begin{array}{r}\sqrt{36\cdot {{x}^{4+2}}}\\\sqrt{36\cdot {{x}^{6}}}\end{array}[/latex]
Look for perfect squares in the radicand.[latex] \sqrt{{{(6)}^{2}}\cdot {{({{x}^{3}})}^{2}}}[/latex]
Rewrite as the product of radicals.[latex] \begin{array}{c}\sqrt{{{(6)}^{2}}}\cdot \sqrt{{{({{x}^{3}})}^{2}}}\\6\cdot {{x}^{3}}\end{array}[/latex]
The answer is [latex]6{{x}^{3}}[/latex].Analysis of the Solution
Even though our answer contained a variable with an odd exponent that was simplified from an even indexed root, we don't need to write our answer with absolute value because we specified before we simplified that [latex] x\ge 0[/latex]. It is important to read the problem very well when you are doing math. Even the smallest statement like [latex] x\ge 0[/latex] can influence the way you write your answer. In our next example, we will multiply two cube roots.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], so [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]
The answer is [latex]\frac{4\sqrt{3}}{5}[/latex].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]
The answer is [latex]\sqrt{3}[/latex].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].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].Licenses & Attributions
CC licensed content, Original
- Multiply Square Roots. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Multiply Cube Roots. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Dividing Radicals without Variables (Basic with no rationalizing). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Dividing Radicals with Variables (Basic with no rationalizing). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: 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 Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Precalculus. Provided by: OpenStax Authored by: Abramson, Jay. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Dwonload fro free at : http://cnx.org/contents/[email protected]:1/Preface.