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
What are Absolute Values and Perfect Square Monomials
Absolute Value: The absolute value is the magnitude of a value regardless of whether it is positive or negative. Mathematically, we find the absolute value by "dropping" the negative if one exists. The solution to an absolute value must always be positive. For example, {eq}|-3| = |3| = 3 {/eq}.
Square Root: The square root is a mathematical operation in which we find the 2nd root of a term.
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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}.
Example 2
Simplify the radical expression making sure to use absolute values where necessary.
{eq}\sqrt{36a^8b^{12}c^{18}} {/eq}
Step 1: First, we simplify the terms of the radical to find the principal root.
{eq}\begin{align*} \sqrt{36a^8b^{12}c^{18}} &= \sqrt{6^2(a^4)^2(b^6)^2(c^9)^2} \\ &= 6a^4b^6c^9 \end{align*} {/eq}
Step 2: Next, for each factor of the monomial solution from the previous step, we evaluate whether it meets the 3 conditions that necessitate the use of absolute value signs.
| Requirement | a^4 | b^6 | c^9 |
|---|---|---|---|
| Even Root Index | Yes | Yes | Yes |
| Even Power Inside | Yes | Yes | Yes |
| Odd Power Outside | No | No | Yes |
Since the factor {eq}c^9 {/eq} satisfies all of the conditions, we must use the absolute values signs around it in the solution.
{eq}\boxed{\sqrt{36a^8b^{12}c^{18}} = 6a^4b^6|c^9|} {/eq}
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