Topics
4. Radical Functions
topic badge
United States of AmericaVirginia
Algebra 2

4.04 Multiply and divide radical expressions

Lesson
Practice

Multiply and divide radical expressions

The same operations that apply to numeric radicals can also be applied to algebraic radical expressions:

  • Multiplication: For radicals with the same index, multiply the coefficients, multiply the radicands, and write under a single radicand before checking to see if the radicand can be simplified further.a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy}, \text{ for }x,y \geq 0

  • Division: For radicals with the same index, divide the coefficients, divide the radicands, and write under a single radicand before checking to see if the radicand can be simplified further.\frac{a\sqrt[n]{x}}{b\sqrt[n]{y}}=\frac{a}{b}\sqrt[n]{\frac{x}{y}}, \text{ for }x \geq 0, y \gt 0, b\neq0

Examples

Example 1

Assuming that each variable represents a non-negative number, fully simplify each expression, writing them as a single radical:

a

3\sqrt[3]{4x^{2}}\cdot 5\sqrt[3]{16x}

Worked Solution
Create a strategy

We can use the product of radicals property to combine the radicals: a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy}Then, multiply the numeric values by one another and multiply the variables together.

Apply the idea
\displaystyle 3\sqrt[3]{4x^{2}}\cdot 5\sqrt[3]{16x}\displaystyle =\displaystyle 15\sqrt[3]{4\left(16\right)x\cdot x^{2}}Use a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy}
\displaystyle =\displaystyle 15\sqrt[3]{64x^{3}}Simplify the products
\displaystyle =\displaystyle 15\sqrt[3]{4^{3}x^{3}}Express 64 as a power of 3
\displaystyle =\displaystyle 15\sqrt[3]{4^{3}}\sqrt[3]{x^{3}}Use \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}
\displaystyle =\displaystyle 60xEvaluate the radicals
Reflect and check

If we chose to separate the numeric and algebraic products in the radicand first, our work to find the solution would be similar to the following:

\displaystyle 3\sqrt[3]{4x^{2}}\cdot 5\sqrt[3]{16x}\displaystyle =\displaystyle 15\sqrt[3]{4}\sqrt[3]{16}\sqrt[3]{x^{2}}\sqrt[3]{x}
\displaystyle =\displaystyle 15\sqrt[3]{64}\sqrt[3]{x^{3}}
\displaystyle =\displaystyle 15\left(4\right)x
\displaystyle =\displaystyle 60x
b

\dfrac{\sqrt{20p^{3}}}{\sqrt{125p^{2}}}

Worked Solution
Create a strategy

We can use the quotient of radicals property to combine the radicals: \frac{\sqrt[n]{ax}}{\sqrt[n]{by}}=\sqrt[n]{\frac{ax}{by}} Then, we can divide the numeric terms and the variables.

Apply the idea
\displaystyle \frac{\sqrt{20p^{3}}}{\sqrt{125p^{2}}}\displaystyle =\displaystyle \sqrt{\frac{20p^{3}}{125p^{2}}}Use \dfrac{\sqrt[n]{ax}}{\sqrt[n]{by}}=\sqrt[n]{\dfrac{ax}{by}}
\displaystyle =\displaystyle \sqrt{\frac{4p}{25}}Simplify the quotient
\displaystyle =\displaystyle \frac{\sqrt{4p}}{\sqrt{25}}Use \sqrt[n]{\dfrac{ax}{by}}=\dfrac{\sqrt[n]{ax}}{\sqrt[n]{by}}
\displaystyle =\displaystyle \frac{2\sqrt{p}}{5}Simplify the radical
Reflect and check

In the provided simplification, we assume that p \gt 0 to ensure the expression under the square root is positive, which allows us to directly divide the terms. However, if p were negative, say p=-2, the expression inside the square root becomes negative, leading us to consider complex numbers.

\displaystyle \frac{\sqrt{20p^{3}}}{\sqrt{125p^{2}}}\displaystyle =\displaystyle \frac{\sqrt{20\left(-2\right)^{3}}}{\sqrt{125\left(-2\right)^{2}}}Substitute p=-2
\displaystyle =\displaystyle \frac{\sqrt{-160}}{\sqrt{500}}Evaluate
\displaystyle =\displaystyle \frac{i\sqrt{160}}{\sqrt{500}}Rewrite using definition of i
\displaystyle =\displaystyle i\sqrt{\frac{160}{500}}Quotient of radicals property
\displaystyle =\displaystyle i\sqrt{\dfrac{8}{25}}Simplify
\displaystyle =\displaystyle \frac{i\sqrt{8}}{\sqrt{25}}Quotient of radicals property
\displaystyle =\displaystyle \frac{2i\sqrt{2}}{5}Simplify

Hence, if y=-2, simplifying the original expression would result in \dfrac{2i\sqrt{2}}{5}, introducing a factor of i due to the square root of a negative number.

The assumption that p is non-negative also excludes p=0, because that would lead to an undefined expression due to the denominator.

Example 2

Fully simplify each of the following expressions, writing them as a single radical. Assume all variables are non-zero.

a

\sqrt[3]{72k^{6}} \cdot 3\sqrt[3]{2k^{3}}

Worked Solution
Create a strategy

We can use the product of radicals property to combine the radicals: a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy}Then, multiply the numeric values by one another and multiply the variables together.

Apply the idea
\displaystyle \sqrt[3]{72k^{6}} \cdot 3\sqrt[3]{2k^{3}}\displaystyle =\displaystyle 3\sqrt[3]{72\cdot 2 \cdot k^{6} \cdot k^{3}}Use a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy}
\displaystyle =\displaystyle 3\sqrt[3]{144\cdot k^{9}}Simplify the products
\displaystyle =\displaystyle 3\sqrt[3]{2^{4}\cdot3^{2}\cdot k^{9}}Express 144 as 2^{4}\cdot3^{2}
\displaystyle =\displaystyle 3\cdot 2 \cdot k^{3} \sqrt[3]{2\cdot 9}Simplify the cube root
\displaystyle =\displaystyle 6k^{3} \sqrt[3]{18}Further simplify
b

\dfrac{\sqrt[3]{64z^{12}}}{\sqrt[3]{z^{4}}}

Worked Solution
Create a strategy

We can use the quotient of radicals property to combine the radicals: \dfrac{\sqrt[n]{ax}}{\sqrt[n]{by}}=\sqrt[n]{\dfrac{ax}{by}} Then, we can divide the numeric terms and the variables.

Apply the idea
\displaystyle \frac{\sqrt[3]{64z^{12}} } {\sqrt[3]{z^{4}}}\displaystyle =\displaystyle \sqrt[3]{\frac{64z^{12}}{z^{4}}} Use \dfrac{\sqrt[n]{ax}}{\sqrt[n]{by}}=\sqrt[n]{\dfrac{ax}{by}}
\displaystyle =\displaystyle \sqrt[3]{64z^{8}} Simplify the expression inside the radical
\displaystyle =\displaystyle 4z^{3}\sqrt[3]{z^{2}} Simplify the cube root

Example 3

Fully simplify this expression, where k \geq 0:\left(7 - 3\sqrt{k}\right)\left(2 + \sqrt{k}\right)

Worked Solution
Create a strategy

We can use the distributive property:\left(a+b\right)\left(c+d\right)=ac+ad+bc+bd and then combine like terms to simplify.

Apply the idea
\displaystyle \left(7 - 3\sqrt{k}\right)\left(2 + \sqrt{k}\right)\displaystyle =\displaystyle 7\left(2\right)+7\sqrt{k}-3\sqrt{k}\left(2\right)-3\sqrt{k}\sqrt{k}Distributive property
\displaystyle =\displaystyle 14+7\sqrt{k}-6\sqrt{k}-3kSimplify the products
\displaystyle =\displaystyle 14+\sqrt{k}-3kCombine like terms
Idea summary

When multiplying and dividing algebraic radicals we can approach this in a couple of different ways:

  • Combine the radicals first, then evaluate the multiplication or division of the radicands

  • Separate the numeric and algebraic products into their own radicals, then simplify and multiply or divide the resulting coefficients and radicals

When multiplying radicals with the same index, multiply the coefficients, multiply the radicands, and write under a single radical.a\sqrt[n]{x}\cdot b\sqrt[n]{y}=ab\sqrt[n]{xy}, \text{ for }x,y \geq 0

When dividing radicals with the same index, divide the coefficients, divide the radicands, and write under a single radical.\dfrac{a\sqrt[n]{x}}{b\sqrt[n]{y}}=\dfrac{a}{b}\sqrt[n]{\dfrac{x}{y}}, \text{ for }x,y \geq 0

Rationalize the denominator

Fractions with radicals in the denominator are not considered to be in a fully simplified form. For these fractions, we can rationalize the denominator, which is a method used to rewrite the expression without radicals in the denominator.

Exploration

For the expression \dfrac{8}{\sqrt{7}}:

  1. What type of number is \sqrt{7}\cdot \sqrt{7}?

  2. Evaluate \dfrac{8}{\sqrt{7}\cdot \sqrt{7}}. Does the result have the same value as the original expression? Use your calculator to verify your answer.

  3. What number can we multiply by that does not change the value of the original number?

  4. Using your answer, what fraction should we multiply\dfrac{8}{\sqrt{7}} by that will not change the value of the expression, but will eliminate the radical from the denominator?

  5. Multiply \dfrac{8}{\sqrt{7}} by your answer to the previous question. Does the result have the same value as the original expression? Use your calculator to verify your answer.

When an expression has only one term in the denominator, we can rationalize the denominator by multiplying the numerator and denominator by the radical in the denominator.

To rationalize the denominator of an expression in the form \dfrac{a}{b\sqrt{n}}, we want to multiply it by the fraction \dfrac{\sqrt{n}}{\sqrt{n}}:

\frac{a}{b\sqrt{n}}\cdot \frac{\sqrt{n}}{\sqrt{n}}=\frac{a\sqrt{n}}{bn}Since we are multiplying the numerator and denominator by the same number, it is the same as multiplying by 1, which does not change the value of the expression.

Examples

Example 4

Express the fraction in simplest form with a rational denominator: \frac{2}{\sqrt[3]{56}}

Worked Solution
Create a strategy

First, check if we can simplify the radical. Then, we will need to rationalize the denominator. To do this, we will need to find a cube root that, when multiplied to the simplified radical, creates a perfect cube in the radicand.

Apply the idea
\displaystyle \frac{2}{\sqrt[3]{56}}\displaystyle =\displaystyle \frac{2}{\sqrt[3]{2\cdot 2\cdot 2\cdot 7}}Find the prime factorization of 56
\displaystyle =\displaystyle \frac{2}{2\sqrt[3]{7}}Simplify the radical
\displaystyle =\displaystyle \frac{1}{\sqrt[3]{7}}Simplify the fraction

When rationalizing the denominator, the goal is for the denominator to become a non-radical expression. This is only possible when the radicand is a perfect cube. If we multiply \sqrt[3]{7} by \sqrt[3]{7^{2}}, this would make the radicand 7^{3} which is a perfect cube.

\displaystyle \frac{1}{1\sqrt[3]{7}}\displaystyle =\displaystyle \frac{1}{\sqrt[3]{7}} \cdot \frac{\sqrt[3]{7^{2}}}{\sqrt[3]{7^{2}}}Rationalize the denominator
\displaystyle =\displaystyle \frac{\sqrt[3]{7^{2}}}{\sqrt[3]{7^{3}}}Evaluate the multiplication
\displaystyle =\displaystyle \frac{\sqrt[3]{49}}{7}Simplify
Reflect and check

If we had multiplied the numerator and denominator by \sqrt[3]{7} when rationalizing the denominator, there would still have been a radical expression in the denominator. \frac{1}{\sqrt[3]{7}} \cdot \frac{\sqrt[3]{7}}{\sqrt[3]{7}}=\frac{\sqrt[3]{7}}{\sqrt[3]{49}}

\sqrt[3]{49} cannot be simplified because it does not contain a perfect cube factor in the radicand. There is still a radical in the denominator which means we need to try rationalizing again.\dfrac{\sqrt[3]{7}}{\sqrt[3]{49}}\cdot \dfrac{\sqrt[3]{7}}{\sqrt[3]{7}}=\dfrac{\sqrt[3]{49}}{\sqrt[3]{343}}

Since 343=7^3, the denominator can now be simplified to a non-radical expression.

When rationalizing denominators that are not square root expressions, we have to think a bit more about what expression to multiply by that would create a perfect n^{\text{th}} root in the denominator.

Example 5

Simplify \dfrac{\sqrt{-39}+\sqrt{6}}{\sqrt{3x}}. All variables are non-negative.

Worked Solution
Create a strategy

Notice that 3 is a factor of both -39 and 6. Because of this, we can simplify the fraction first before rationalizing.

Apply the idea
\displaystyle \frac{\sqrt{-39}+\sqrt{6}}{\sqrt{3x}}\displaystyle =\displaystyle \frac{\sqrt{-39}}{\sqrt{3x}}+\frac{\sqrt{6}}{\sqrt{3x}}Separate into two fractions
\displaystyle =\displaystyle \frac{i\sqrt{39}}{\sqrt{3x}}+\frac{\sqrt{6}}{\sqrt{3x}}Rewrite using the definition of i
\displaystyle =\displaystyle i\sqrt{\dfrac{39}{3x}}+\sqrt{\dfrac{6}{3x}}Use the quotient of radicals rule
\displaystyle =\displaystyle i\sqrt{\frac{13}{x}}+\sqrt{\frac{2}{x}}Evaluate the division
\displaystyle =\displaystyle i\frac{\sqrt{13}}{\sqrt{x}}\cdot\frac{\sqrt{x}}{\sqrt{x}}+\frac{\sqrt{2}}{\sqrt{x}}\cdot \frac{\sqrt{x}}{\sqrt{x}}Rationalize the denominators
\displaystyle =\displaystyle i\frac{\sqrt{13x}}{x}+\frac{\sqrt{2x}}{x}Evaluate the multiplication
Reflect and check

Rationalizing the denominator first would have given us the same result:

\displaystyle \frac{\sqrt{-39}+\sqrt{6}}{\sqrt{3x}}\displaystyle =\displaystyle \frac{i\sqrt{39}+\sqrt{6}}{\sqrt{3x}}Rewrite using the definition of i
\displaystyle =\displaystyle \frac{i\sqrt{39}+\sqrt{6}}{\sqrt{3x}}\cdot \frac{\sqrt{3x}}{\sqrt{3x}}Rationalize the denominator
\displaystyle =\displaystyle \frac{i\sqrt{117x}+\sqrt{18x}}{3x}Evaluate the multiplication
\displaystyle =\displaystyle \frac{3i\sqrt{13x}+3\sqrt{2x}}{3x}Simplify the radicands
\displaystyle =\displaystyle \frac{i\sqrt{13x}+\sqrt{2x}}{x}Evaluate the division

Example 6

For the expression \dfrac{4g^{\frac{1}{2}}}{8h^{\frac{3}{2}}}:

a

Convert to a radical expression.

Worked Solution
Create a strategy

To convert the given expression to a radical form, we'll replace the fractional exponents with radicals. Remember, a^{\frac{m}{n}} = \sqrt[n]{a^{m}}.

Apply the idea

Starting with the original expression:

\frac{4g^{\frac{1}{2}}}{8h^{\frac{3}{2}}}

We can rewrite it as:

\frac{4\sqrt{g}}{8\sqrt{h^{3}}}

Reflect and check

This conversion allows us to more clearly see the radical nature of the expression, which can be useful for simplification and evaluation steps that follow.

b

Evaluate the quotient. Simplify fully, including rationalizing the denominator.

Worked Solution
Create a strategy

To simplify the expression, we first reduce the coefficients and divide the radicands, if possible. To rationalize the denominator, we need to multiply the numerator and denominator by a radical that combines with \sqrt{h^{3}} to create perfect square radicand.

Apply the idea

Reducing the coefficient, we get:

\frac{\sqrt{g}}{2\sqrt{h^{3}}}

To rationalize the denominator, we can multiply by \dfrac{\sqrt{h}}{\sqrt{h}}. This will create a radicand of h^{4}, which is a perfect square: \left(h^{2}\right)^{2}.

\displaystyle \frac{\sqrt{g}}{2\sqrt{h^3}}\displaystyle =\displaystyle \frac{\sqrt{g} \cdot \sqrt{h}}{2\sqrt{h^{3}} \cdot \sqrt{h}}
\displaystyle =\displaystyle \frac{\sqrt{gh}}{2\sqrt{h^4}}
\displaystyle =\displaystyle \frac{\sqrt{gh}}{2h^{2}}
Reflect and check

Rather than using radicals to simplify, we could have used exponent laws to simplify and rationalize the expression.

\displaystyle \frac{4g^{\frac{1}{2}}}{8h^{\frac{3}{2}}}\displaystyle =\displaystyle \frac{g^{\frac{1}{2}}}{2h^{\frac{3}{2}}}Divide the coefficients
\displaystyle =\displaystyle \frac{g^{\frac{1}{2}}}{2h^{\frac{3}{2}}}\cdot \frac{h^{\frac{1}{2}}}{h^{\frac{1}{2}}}Rationalize the denominator
\displaystyle =\displaystyle \frac{g^{\frac{1}{2}}h^{\frac{1}{2}}}{2h^{\frac{4}{2}}}Product of powers property
\displaystyle =\displaystyle \frac{g^{\frac{1}{2}}h^{\frac{1}{2}}}{2h^2}Simplify the exponent in the denominator
\displaystyle =\displaystyle \frac{\left(gh\right)^{\frac{1}{2}}}{2h^2}Power of a product property

Converting the numerator to a radical expression, we can see this answer is the same as the one we found previously.

Idea summary

To rationalize the denominator of an expression in the form \dfrac{a}{b\sqrt{n}}, we want to multiply it by the fraction \dfrac{\sqrt{n}}{\sqrt{n}}:

\dfrac{a}{b\sqrt{n}}\cdot \dfrac{\sqrt{n}}{\sqrt{n}}=\dfrac{a\sqrt{n}}{bn}

To rationalize a denominator containing an nth root expression, we multiply the numerator and denominator by a radical that will create a perfect nth power in the denominator's radicand.

If possible, we should simplify the radicals before rationalizing.

Outcomes

A2.EO.2

The student will perform operations on and simplify radical expressions.

A2.EO.2b

Add, subtract, multiply, and divide radical expressions that include numeric and algebraic radicands, simplifying the result. Simplification may include rationalizing the denominator.

What is Mathspace

About Mathspace