Simplify radical expressions using rational exponents and the laws of exponents
Define x2=∣x∣ and apply it when simplifying radical expressions
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.
To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the Product Raised to a Power Rule from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written (ab)x=ax⋅bx. So, for example, you can use the rule to rewrite (3x)2 as 32⋅x2=9⋅x2=9x2.
Now instead of using the exponent 2, use the exponent 21. The exponent is distributed in the same way.
(3x)21=321⋅x21
And since you know that raising a number to the 21 power is the same as taking the square root of that number, you can also write it this way.
3x=3⋅x
Look at that—you can think of any number underneath a radical as the product of separate factors, each underneath its own radical.
A Product Raised to a Power Rule or sometimes called The Square Root of a Product Rule
For any real numbers a and b, ab=a⋅b.
For example: 100=10⋅10, and 75=25⋅3
This rule is important because it helps you think of one radical as the product of multiple radicals. If you can identify perfect squares within a radical, as with (2⋅2)(2⋅2)(3⋅3), you can rewrite the expression as the product of multiple perfect squares: 22⋅22⋅32.
The square root of a product rule will help us simplify roots that are not perfect as is shown the following example.
Example
Simplify. 63
Answer:
63 is not a perfect square so we can use the square root of a product rule to simplify any factors that are perfect squares.
Factor 63 into 7 and 9.
7⋅99 is a perfect square, 9=32, therefore we can rewrite the radicand.
7⋅32
Using the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical.
7⋅32
Take the square root of 32.
7⋅3
Rearrange factors so the integer appears before the radical and then multiply. This is done so that it is clear that only the 7 is under the radical, not the 3.
3⋅7
The answer is 37.
The final answer 37 may look a bit odd, but it is in simplified form. You can read this as “three radical seven” or “three times the square root of seven.”
The following video shows more examples of how to simplify square roots that do not have perfect square radicands.
https://youtu.be/oRd7aBCsmfU
Before we move on to simplifying more complex radicals with variables, we need to learn about an important behavior of square roots with variables in the radicand.
Consider the expression x2. This looks like it should be equal to x, right? 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! However, in all cases x2=∣x∣. You need to consider this fact when simplifying radicals with an even index 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 an even power, remember that x2=∣x∣.
Examples: 9x2=3∣x∣, and 16x2y2=4∣xy∣
We will combine this with the square root of a product rule in our next example to simplify an expression with three variables in the radicand.
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.
∣ac∣b2ab
Analysis of the Solution
Why did we not write b2 as ∣b2∣? Because when you square a number, you will always get a positive result, so the principal square root of (b2)2 will always be non-negative. One tip for knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. If the exponent is odd - including 1 - add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples.
In the following video, you will see more examples of how to simplify radical expressions with variables.
https://youtu.be/q7LqsKPoAKo
We will show another example where the simplified expression contains variables with both odd and even powers.
Example
Simplify. 9x6y4
Answer:
Factor to find identical pairs.
3⋅3⋅x3⋅x3⋅y2⋅y2
Rewrite the pairs as perfect squares.
32⋅(x3)2⋅(y2)2
Separate into individual radicals.
32⋅(x3)2⋅(y2)2
Simplify.
3x3y2
Because x has an odd power, we will add the absolute value for our final solution.
3∣x3∣y2
In our next example, we will start with an expression written with a rational exponent. You will see that you can use a similar process - factoring and sorting terms into squares - to simplify this expression.
Example
Simplify. (36x4)21
Answer:
Rewrite the expression with the fractional exponent as a radical.
36x4
Find the square root of both the coefficient and the variable.
62⋅x462⋅x462⋅(x2)26⋅x2
The answer is 6x2.
Here is one more example with perfect squares.
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
Simplify Cube Roots
We can use the same techniques we have used for simplifying square roots to simplify higher order roots. For example, to simplify a cube root, the goal is to find factors under the radical that are perfect cubes so that you can take their cube root. We no longer need to be concerned about whether we have identified the principal root since we are now finding cube roots. Focus on finding identical trios of factors as you simplify.
Example
Simplify. 340m5
Answer:
Factor 40 into prime factors.
35⋅2⋅2⋅2⋅m5
Since you are looking for the cube root, you need to find factors that appear 3 times under the radical. Rewrite 2⋅2⋅2 as 23.
323⋅5⋅m5
Rewrite m5 as m3⋅m2.
323⋅5⋅m3⋅m2
Rewrite the expression as a product of multiple radicals.
323⋅35⋅3m3⋅3m2
Simplify and multiply.
2⋅35⋅m⋅3m2
The answer is 2m35m2.
Remember that you can take the cube root of a negative expression. In the next example, we will simplify a cube root with a negative radicand.
Example
Simplify. 3−27x4y3
Answer:
Factor the expression into cubes.
Separate the cubed factors into individual radicals.
3−1⋅27⋅x4⋅y33(−1)3⋅(3)3⋅x3⋅x⋅y33(−1)3⋅3(3)3⋅3x3⋅3x⋅3y3
Simplify the cube roots.
−1⋅3⋅x⋅y⋅3x
The answer is 3−27x4y3=−3xy3x.
You could check your answer by performing the inverse operation. If you are right, when you cube −3xy3x you should get −27x4y3.
(−3xy3x)(−3xy3x)(−3xy3x)−3⋅−3⋅−3⋅x⋅x⋅x⋅y⋅y⋅y⋅3x⋅3x⋅3x−27⋅x3⋅y3⋅3x3−27x3y3⋅x−27x4y3
You can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.
Example
Simplify. 3−24a5
Answer:
Factor −24 to find perfect cubes. Here, −1 and 8 are the perfect cubes.
3−1⋅8⋅3⋅a5
Factor variables. You are looking for cube exponents, so you factor a5 into a3 and a2.
3(−1)3⋅23⋅3⋅a3⋅a2
Separate the factors into individual radicals.
3(−1)3⋅323⋅3a3⋅33⋅a2
Simplify, using the property 3x3=x.
−1⋅2⋅a⋅33⋅a2
This is the simplest form of this expression; all cubes have been pulled out of the radical expression.
−2a33a2
In the following video, we show more examples of simplifying cube roots.
https://youtu.be/BtJruOpmHCE
Simplifying Fourth Roots
Now let us move to simplifying fourth degree roots. No matter what root you are simplifying, the same idea applies: find cubes for cube roots, powers of four for fourth roots, etc. Recall that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to apply an absolute value.
Example
Simplify. 481x8y3
Answer:
Rewrite the expression.
481⋅4x8⋅4y3
Factor each radicand.
43⋅3⋅3⋅3⋅4x2⋅x2⋅x2⋅x2⋅4y3
Simplify.
434⋅4(x2)4⋅4y33⋅x2⋅4y3
The answer is 481x8y3=3x24y3.
An alternative method to factoring is to rewrite the expression with rational exponents, then use the rules of exponents to simplify. You may find that you prefer one method over the other. Either way, it is nice to have options. We will show the last example again, using this idea.
Example
Simplify. 481x8y3
Answer:
Rewrite the radical using rational exponents.
(81x8y3)41
Use the rules of exponents to simplify the expression.
8141⋅x48⋅y43(3⋅3⋅3⋅3)41x2y43(34)41x2y433x2y43
Change the expression with the rational exponent back to radical form.
3x24y3
In the following video, we show another example of how to simplify a fourth and fifth root.
https://youtu.be/op2LEb0YRyw
For our last example, we will simplify a more complicated expression, c38b410b2c2.This expression has two variables, a fraction, and a radical. Let us take it step-by-step and see if using fractional exponents can help us simplify it.
We will start by simplifying the denominator since this is where the radical sign is located. Recall that an exponent in the denominator of a fraction can be rewritten as a negative exponent.
Example
Simplify. c38b410b2c2
Answer:
Separate the factors in the denominator.
c⋅38⋅3b410b2c2
Take the cube root of 8, which is 2.
c⋅2⋅3b410b2c2
Rewrite the radical using a fractional exponent.
c⋅2⋅b3410b2c2
Rewrite the fraction as a series of factors in order to cancel factors (see next step).
210⋅cc2⋅b34b2
Simplify the constant and c factors.
5⋅c⋅b34b2
Use the rule of negative exponents, n-x=nx1, to rewrite b341 as b−34.
5cb2b−34
Combine the b factors by adding the exponents.
5cb32
Change the expression with the fractional exponent back to radical form. By convention, an expression is not usually considered simplified if it has a fractional exponent or a radical in the denominator.
5c3b2
Well, that took a while, but you did it. You applied what you know about fractional exponents, negative exponents, and the rules of exponents to simplify the expression.
In our last video, we show how to use rational exponents to simplify radical expressions.
https://youtu.be/CfxhFRHUq_M
Summary
A radical expression is a mathematical way of representing the nth root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property nxn=x if n is odd and nxn=∣x∣ if n is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions.
The steps to consider when simplifying a radical are outlined below.
Simplifying a radical
When working with exponents and radicals:
If n is odd, nxn=x.
If n is even, nxn=∣x∣. (The absolute value accounts for the fact that if x is negative and raised to an even power, that number will be positive, as will the nth principal root of that number.)
Licenses & Attributions
CC licensed content, Original
Simplify Square Roots (Not Perfect Square Radicands).Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: 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.
Simplify Cube Roots (Not Perfect Cube Radicands).Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
Simplify Nth Roots with Variables.Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
Simplify Radicals Using Rational Exponents.Authored by: James Sousa (Mathispower4u.com) for Lumen Learning.License: CC BY: Attribution.
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Precalculus.Provided by: OpenStaxAuthored by: Abramson, Jay.Located at: https://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface.License: CC BY: Attribution. License terms: Download for free at : http://cnx.org/contents/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1/Preface.
Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program.Provided by: Monterey Institute of TechnologyLocated at: https://www.nroc.org/.License: CC BY: Attribution.