Recognize that by definition x2 is always nonnegative
Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as16, to quite complicated, as in 3250x4y. Using factoring, you can simplify these radical expressions, too.
Radical
Simplifying Square Roots
Radical expressions will sometimes include variables as well as numbers. Consider the expression 9x6. Simplifying a radical expression with variables is not as straightforward as the examples we have already shown with integers.
Consider the expression x2. This looks like it should be equal to x, right? Let’s test some values for x and see what happens.
In the chart below, look along each row and determine whether the value of x is the same as the value of x2. Where are they equal? Where are they not equal?
After doing that for each row, look again and determine whether the value of x2 is the same as the value of ∣x∣.
x
x2
x2
∣x∣
−5
25
5
5
−2
4
2
2
0
0
0
0
6
36
6
6
10
100
10
10
Notice—in cases where x is a negative number, x2=x! (This happens because the process of squaring the number loses the negative sign, since a negative times a negative is a positive.) However, in all cases x2=∣x∣. You need to consider this fact when simplifying radicals that contain variables, because by definition x2 is always nonnegative.
Taking the Square Root of a Radical Expression
When finding the square root of an expression that contains variables raised to a power, consider that x2=∣x∣.
Examples: 9x2=3∣x∣, and 16x2y2=4∣xy∣
Let’s try it.
The goal is to find factors under the radical that are perfect squares so that you can take their square root.
Example
Simplify. 9x6
Answer: Factor to find identical pairs.
3⋅3⋅x3⋅x3
Rewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify x6 into a square: x3232⋅(x3)2
Separate into individual radicals.
32⋅(x3)2
Simplify, using the rule that x2=∣x∣.
3x3
Answer
9x6=3x3
Variable factors with even exponents can be written as squares. In the example above, x6=x3⋅x3=x32 and
y4=y2⋅y2=(∣y2)2.
Try It
[ohm_question]189413[/ohm_question]
Let’s try to simplify another radical expression.
Example
Simplify. 100x2y4
Answer: Separate factors; look for squared numbers and variables. Factor 100 into 10⋅10.
10⋅10⋅x2⋅y4
Factor y4 into (y2)2.
10⋅10⋅x2⋅(y2)2
Separate the squared factors into individual radicals.
102⋅x2⋅(y2)2
Take the square root of each radical . Since you do not know whether x is positive or negative, use ∣x∣ to account for both possibilities, thereby guaranteeing that your answer will be positive.
10⋅∣x∣⋅y2
Simplify and multiply.
10∣x∣y2
Answer
100x2y4=10∣x∣y2
You can check your answer by squaring it to be sure it equals 100x2y4.
Example
Simplify. 49x10y8
Answer: Look for squared numbers and variables. Factor 49 into 7⋅7, x10 into x5⋅x5, and y8 into y4⋅y4.
7⋅7⋅x5⋅x5⋅y4⋅y4
Rewrite the pairs as squares.
72⋅(x5)2⋅(y4)2
Separate the squared factors into individual radicals.
72⋅(x5)2⋅(y4)2
Take the square root of each radical using the rule that x2=∣x∣.
7⋅x5⋅y4
Multiply.
7x5y4
Answer
49x10y8=7x5y4
You find that the square root of 49x10y8 is 7x5y4. In order to check this calculation, you could square 7x5y4, hoping to arrive at 49x10y8. And, in fact, you would get this expression if you evaluated (7x5y4)2.
In the video that follows we show several examples of simplifying radicals with variables.
https://youtu.be/q7LqsKPoAKo
Example
Simplify. a3b5c2
Answer: Factor to find variables with even exponents.
a2⋅a⋅b4⋅b⋅c2
Rewrite b4 as (b2)2.
a2⋅a⋅(b2)2⋅b⋅c2
Separate the squared factors into individual radicals.
a2⋅(b2)2⋅c2⋅a⋅b
Take the square root of each radical. Remember that a2=∣a∣.
∣a∣⋅b2⋅∣c∣⋅a⋅b
Simplify and multiply. The entire quantity ab2c can be enclosed in the absolute value sign because b2 will be positive anyway.
ab2cab
Answer
a3b5c2=ab2cab
In the next section, we will explore cube roots, and use the methods we have shown here to simplify them. Cube roots are unique from square roots in that it is possible to have a negative number under the root, such as 3−125.
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Screenshot: radical.Provided by: Lumen LearningLicense: CC BY: Attribution.
Revision and Adaptation.Provided by: Lumen LearningLicense: CC BY: Attribution.
Simplify Square Roots with Variables.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 EducationLocated at: https://www.nroc.org/.License: CC BY: Attribution.