# Using Absolute Value to Simplify Square Roots of Perfect Square Monomials

Kayla Zeliff
• Instructors
Kayla Zeliff

Kayla has a Bachelor’s in Mathematics and a Master’s in Mechanical Engineering. She spent the early portion of her career as a mathematical researcher in the fields of cyber security and machine learning. Currently, she works as a mathematical content developer creating lessons for elementary through college students.

## How to Use Absolute Value to Simplify Square Roots of Perfect Square Monomials

Step 1: Compute the square root of the monomial using the rule {eq}\sqrt[n]{a^n} = a {/eq}.

Step 2: Working variable by variable, apply an absolute variable sign if all 3 of the following conditions are met:

• The root index is even
• The variable has an even power inside
• The variable contains an odd power outside

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Parts of a Radical: A radical expression contains 3 components. The radical itself, the index, and the radicand. A radical expression, commonly referred to as the {eq}n {/eq}th root, is written as {eq}\sqrt[n]{a} {/eq}. The integer {eq}n {/eq} is the index and the value {eq}a {/eq} is the radicand.

Principal Root: The principal root of an nth root is the positive solution to the radical. For example, given {eq}\sqrt{4} = \pm 2 {/eq}, the principal root is the positive, {eq}2 {/eq}.

Let's practice using the absolute value to simplify the square roots of perfect squares by working through 2 examples in detail.

## Examples for Using Absolute Value to Simplify Square Roots of Perfect Square Monomials

Example 1

Simplify the radical expression making sure to use absolute values where necessary.

{eq}\sqrt{64x^{10}y^4} {/eq}

Step 1: First, we compute the square root of the radicand. Since we are taking the square root, we know the index is {eq}2 {/eq}.

{eq}\begin{align*} \sqrt{64x^{10}y^4} &=\sqrt{8^2(x^5)^2(y^2)^2} \\ &= 8x^5y^2 \end{align*} {/eq}

Step 2: Now that we have the principal root of the perfect square monomial, we must look at each of the variable factors of the solution and determine whether we must apply the absolute value.

Let's start with the first requirement- the root index is even. Since we are taking the square root of the monomial, the index is {eq}n = 2 {/eq} and {eq}2 {/eq} is an even number. Therefore, both factors of the solution meet the first qualification.

The second requirement is that the power of the radicand must be even. In the original problem, the radicand is {eq}64x^{10}y^4 {/eq}. The exponents of the variables are {eq}10 {/eq} and {eq}4 {/eq}. These are both even numbers, so the second requirement for absolute value signs has been satisfied.

Finally, we explore the exponents of the solution. The factor {eq}x^5 {/eq} has an odd exponent and the factor {eq}y^2 {/eq} has an even exponent.

We use a table to summarize the requirements for each variable term, {eq}x^5 {/eq} and {eq}y^2 {/eq}.

Requirement x^5 y^2
Even Root Index Yes Yes
Even Power Inside Yes Yes
Odd Power Outside Yes No

From the table, we determine the factor {eq}x^5 {/eq} requires the absolute value, but the term {eq}y^2 {/eq} does not. Therefore, {eq}\boxed{\sqrt{64x^{10}y^4} = 64|x^5|y^2} {/eq}.

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