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Study Guides > Intermediate Algebra

Simplify Radical Expressions

Learning Outcomes

  • Simplify radical expressions using factoring
  • Simplify radical expressions using rational exponents and the laws of exponents
  • Define [latex]\sqrt{x^2}=|x|[/latex] 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 as[latex] \sqrt{16}[/latex], to quite complicated, as in [latex] \sqrt[3]{250{{x}^{4}}y}[/latex]. 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 [latex]\left(ab\right)^{x}=a^{x}\cdot{b}^{x}[/latex]. So, for example, you can use the rule to rewrite [latex] {{\left( 3x \right)}^{2}}[/latex] as [latex] {{3}^{2}}\cdot {{x}^{2}}=9\cdot {{x}^{2}}=9{{x}^{2}}[/latex]. Now instead of using the exponent [latex]2[/latex], use the exponent [latex] \frac{1}{2}[/latex]. The exponent is distributed in the same way.

[latex] {{\left( 3x \right)}^{\frac{1}{2}}}={{3}^{\frac{1}{2}}}\cdot {{x}^{\frac{1}{2}}}[/latex]

And since you know that raising a number to the [latex] \frac{1}{2}[/latex] power is the same as taking the square root of that number, you can also write it this way.

[latex] \sqrt{3x}=\sqrt{3}\cdot \sqrt{x}[/latex]

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, [latex] \sqrt{ab}=\sqrt{a}\cdot \sqrt{b}[/latex]. For example: [latex] \sqrt{100}=\sqrt{10}\cdot \sqrt{10}[/latex], and [latex] \sqrt{75}=\sqrt{25}\cdot \sqrt{3}[/latex]
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 [latex] \sqrt{(2\cdot 2)(2\cdot 2)(3\cdot 3})[/latex], you can rewrite the expression as the product of multiple perfect squares: [latex] \sqrt{{{2}^{2}}}\cdot \sqrt{{{2}^{2}}}\cdot \sqrt{{{3}^{2}}}[/latex].

The square root of a product rule will help us simplify roots that are not perfect as is shown the following example.

Example

Simplify. [latex] \sqrt{63}[/latex]

Answer: [latex]63[/latex] 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 [latex]63[/latex] into [latex]7[/latex] and [latex]9[/latex]. [latex-display] \sqrt{7\cdot 9}[/latex-display] [latex]9[/latex] is a perfect square, [latex]9=3^2[/latex], therefore we can rewrite the radicand. [latex-display] \sqrt{7\cdot {{3}^{2}}}[/latex-display] Using the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical. [latex-display] \sqrt{7}\cdot \sqrt{{{3}^{2}}}[/latex-display] Take the square root of [latex]3^{2}[/latex]. [latex-display] \sqrt{7}\cdot 3[/latex-display] Rearrange factors so the integer appears before the radical and then multiply. This is done so that it is clear that only the [latex]7[/latex] is under the radical, not the [latex]3[/latex]. [latex-display] 3\cdot \sqrt{7}[/latex-display] The answer is [latex]3\sqrt{7}[/latex].

The final answer [latex] 3\sqrt{7}[/latex] 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 [latex] \sqrt{{{x}^{2}}}[/latex]. 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 [latex] \sqrt{{{x}^{2}}}[/latex]. Where are they equal? Where are they not equal? After doing that for each row, look again and determine whether the value of [latex] \sqrt{{{x}^{2}}}[/latex] is the same as the value of [latex]\left|x\right|[/latex].
[latex]x[/latex] [latex]x^{2}[/latex] [latex]\sqrt{x^{2}}[/latex] [latex]\left|x\right|[/latex]
[latex]−5[/latex] [latex]25[/latex] [latex]5[/latex] [latex]5[/latex]
[latex]−2[/latex] [latex]4[/latex] [latex]2[/latex] [latex]2[/latex]
[latex]0[/latex] [latex]0[/latex] [latex]0[/latex] [latex]0[/latex]
[latex]6[/latex] [latex]36[/latex] [latex]6[/latex] [latex]6[/latex]
[latex]10[/latex] [latex]100[/latex] [latex]10[/latex] [latex]10[/latex]
Notice—in cases where x is a negative number, [latex]\sqrt{x^{2}}\neq{x}[/latex]! However, in all cases [latex]\sqrt{x^{2}}=\left|x\right|[/latex]. You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition [latex]\sqrt{x^{2}}[/latex] 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 [latex]\sqrt{x^{2}}=\left|x\right|[/latex]. Examples: [latex]\sqrt{9x^{2}}=3\left|x\right|[/latex], and [latex]\sqrt{16{{x}^{2}}{{y}^{2}}}=4\left|xy\right|[/latex]
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. [latex] \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[/latex]

Answer: Factor to find variables with even exponents. [latex-display] \sqrt{{{a}^{2}}\cdot a\cdot {{b}^{4}}\cdot b\cdot {{c}^{2}}}[/latex-display] Rewrite [latex]b^{4}[/latex] as [latex]\left(b^{2}\right)^{2}[/latex]. [latex-display] \sqrt{{{a}^{2}}\cdot a\cdot {{({{b}^{2}})}^{2}}\cdot b\cdot {{c}^{2}}}[/latex-display] Separate the squared factors into individual radicals. [latex-display] \sqrt{{{a}^{2}}}\cdot \sqrt{{{({{b}^{2}})}^{2}}}\cdot \sqrt{{{c}^{2}}}\cdot \sqrt{a\cdot b}[/latex-display] Take the square root of each radical. Remember that [latex] \sqrt{{{a}^{2}}}=\left| a \right|[/latex]. [latex-display] \left| a \right|\cdot {{b}^{2}}\cdot \left|{c}\right|\cdot \sqrt{a\cdot b}[/latex-display] Simplify and multiply. [latex-display] \left| ac \right|{{b}^{2}}\sqrt{ab}[/latex-display]

Analysis of the Solution

Why did we not write [latex]b^2[/latex] as [latex]|b^2|[/latex]?  Because when you square a number, you will always get a positive result, so the principal square root of [latex]\left(b^2\right)^2[/latex] 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 [latex]1[/latex] - 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. [latex] \sqrt{9{{x}^{6}}{{y}^{4}}}[/latex]

Answer: Factor to find identical pairs.

[latex] \sqrt{3\cdot 3\cdot {{x}^{3}}\cdot {{x}^{3}}\cdot {{y}^{2}}\cdot {{y}^{2}}}[/latex]

Rewrite the pairs as perfect squares.

[latex] \sqrt{{{3}^{2}}\cdot {{\left( {{x}^{3}} \right)}^{2}}\cdot {{\left( {{y}^{2}} \right)}^{2}}}[/latex]

Separate into individual radicals.

[latex] \sqrt{{{3}^{2}}}\cdot \sqrt{{{\left( {{x}^{3}} \right)}^{2}}}\cdot \sqrt{{{\left( {{y}^{2}} \right)}^{2}}}[/latex]

Simplify.

[latex] 3{{x}^{3}}{{y}^{2}}[/latex]

Because x has an odd power, we will add the absolute value for our final solution.

[latex] 3|{{x}^{3}}|{{y}^{2}}[/latex]

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. [latex] {{(36{{x}^{4}})}^{\frac{1}{2}}}[/latex]

Answer: Rewrite the expression with the fractional exponent as a radical.

[latex] \sqrt{36{{x}^{4}}}[/latex]

Find the square root of both the coefficient and the variable.

[latex]\begin{array}{r} \sqrt{{{6}^{2}}\cdot {{x}^{4}}}\\\sqrt{{{6}^{2}}}\cdot \sqrt{{{x}^{4}}}\\\sqrt{{{6}^{2}}}\cdot \sqrt{{{({{x}^{2}})}^{2}}}\\6\cdot{x}^{2}\end{array}[/latex]

The answer is [latex]6{{x}^{2}}[/latex].

Here is one more example with perfect squares.

Example

Simplify. [latex] \sqrt{49{{x}^{10}}{{y}^{8}}}[/latex]

Answer: Look for squared numbers and variables. Factor [latex]49[/latex] into [latex]7\cdot7[/latex], [latex]x^{10}[/latex] into [latex]x^{5}\cdot{x}^{5}[/latex], and [latex]y^{8}[/latex] into [latex]y^{4}\cdot{y}^{4}[/latex].

[latex] \sqrt{7\cdot 7\cdot {{x}^{5}}\cdot {{x}^{5}}\cdot {{y}^{4}}\cdot {{y}^{4}}}[/latex]

Rewrite the pairs as squares.

[latex] \sqrt{{{7}^{2}}\cdot {{({{x}^{5}})}^{2}}\cdot {{({{y}^{4}})}^{2}}}[/latex]

Separate the squared factors into individual radicals.

[latex] \sqrt{{{7}^{2}}}\cdot \sqrt{{{({{x}^{5}})}^{2}}}\cdot \sqrt{{{({{y}^{4}})}^{2}}}[/latex]

Take the square root of each radical using the rule that [latex] \sqrt{{{x}^{2}}}=x[/latex].

[latex] 7\cdot {{x}^{5}}\cdot {{y}^{4}}[/latex]

Multiply.

[latex] 7{{x}^{5}}{{y}^{4}}[/latex]

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. [latex] \sqrt[3]{40{{m}^{5}}}[/latex]

Answer: Factor [latex]40[/latex] into prime factors. [latex-display] \sqrt[3]{5\cdot 2\cdot 2\cdot 2\cdot {{m}^{5}}}[/latex-display] Since you are looking for the cube root, you need to find factors that appear [latex]3[/latex] times under the radical. Rewrite [latex] 2\cdot 2\cdot 2[/latex] as [latex] {{2}^{3}}[/latex]. [latex-display] \sqrt[3]{{{2}^{3}}\cdot 5\cdot {{m}^{5}}}[/latex-display] Rewrite [latex] {{m}^{5}}[/latex] as [latex] {{m}^{3}}\cdot {{m}^{2}}[/latex]. [latex-display] \sqrt[3]{{{2}^{3}}\cdot 5\cdot {{m}^{3}}\cdot {{m}^{2}}}[/latex-display] Rewrite the expression as a product of multiple radicals. [latex-display] \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{5}\cdot \sqrt[3]{{{m}^{3}}}\cdot \sqrt[3]{{{m}^{2}}}[/latex-display] Simplify and multiply. [latex-display] 2\cdot \sqrt[3]{5}\cdot m\cdot \sqrt[3]{{{m}^{2}}}[/latex-display] The answer is [latex]2m\sqrt[3]{5{{m}^{2}}}[/latex].

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. [latex] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[/latex]

Answer: Factor the expression into cubes. Separate the cubed factors into individual radicals. [latex-display]\begin{array}{r}\sqrt[3]{-1\cdot 27\cdot {{x}^{4}}\cdot {{y}^{3}}}\\\sqrt[3]{{{(-1)}^{3}}\cdot {{(3)}^{3}}\cdot {{x}^{3}}\cdot x\cdot {{y}^{3}}}\\\sqrt[3]{{{(-1)}^{3}}}\cdot \sqrt[3]{{{(3)}^{3}}}\cdot \sqrt[3]{{{x}^{3}}}\cdot \sqrt[3]{x}\cdot \sqrt[3]{{{y}^{3}}}\end{array}[/latex-display] Simplify the cube roots. [latex-display] -1\cdot 3\cdot x\cdot y\cdot \sqrt[3]{x}[/latex-display] The answer is [latex] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\sqrt[3]{x}[/latex].

You could check your answer by performing the inverse operation. If you are right, when you cube [latex] -3xy\sqrt[3]{x}[/latex] you should get [latex] -27{{x}^{4}}{{y}^{3}}[/latex]. [latex-display] \begin{array}{l}\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\left( -3xy\sqrt[3]{x} \right)\\-3\cdot -3\cdot -3\cdot x\cdot x\cdot x\cdot y\cdot y\cdot y\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\cdot \sqrt[3]{x}\\-27\cdot {{x}^{3}}\cdot {{y}^{3}}\cdot \sqrt[3]{{{x}^{3}}}\\-27{{x}^{3}}{{y}^{3}}\cdot x\\-27{{x}^{4}}{{y}^{3}}\end{array}[/latex-display] You can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.

Example

Simplify. [latex] \sqrt[3]{-24{{a}^{5}}}[/latex]

Answer: Factor [latex]−24[/latex] to find perfect cubes. Here, [latex]−1[/latex] and [latex]8[/latex] are the perfect cubes.

[latex] \sqrt[3]{-1\cdot 8\cdot 3\cdot {{a}^{5}}}[/latex]

Factor variables. You are looking for cube exponents, so you factor [latex]a^{5}[/latex] into [latex]a^{3}[/latex] and [latex]a^{2}[/latex].

[latex] \sqrt[3]{{{(-1)}^{3}}\cdot {{2}^{3}}\cdot 3\cdot {{a}^{3}}\cdot {{a}^{2}}}[/latex]

Separate the factors into individual radicals.

[latex] \sqrt[3]{{{(-1)}^{3}}}\cdot \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{{{a}^{3}}}\cdot \sqrt[3]{3\cdot {{a}^{2}}}[/latex]

Simplify, using the property [latex] \sqrt[3]{{{x}^{3}}}=x[/latex]. 

[latex] -1\cdot 2\cdot a\cdot \sqrt[3]{3\cdot {{a}^{2}}}[/latex]

This is the simplest form of this expression; all cubes have been pulled out of the radical expression.

[latex] -2a\sqrt[3]{3{{a}^{2}}}[/latex]

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. [latex] \sqrt[4]{81{{x}^{8}}{{y}^{3}}}[/latex]

Answer: Rewrite the expression. [latex-display] \sqrt[4]{81}\cdot \sqrt[4]{{{x}^{8}}}\cdot \sqrt[4]{{{y}^{3}}}[/latex-display] Factor each radicand. [latex-display] \sqrt[4]{3\cdot 3\cdot 3\cdot 3}\cdot \sqrt[4]{{{x}^{2}}\cdot {{x}^{2}}\cdot {{x}^{2}}\cdot {{x}^{2}}}\cdot \sqrt[4]{{{y}^{3}}}[/latex-display] Simplify. [latex-display]\begin{array}{r}\sqrt[4]{{{3}^{4}}}\cdot \sqrt[4]{{{({{x}^{2}})}^{4}}}\cdot \sqrt[4]{{{y}^{3}}}\\3\cdot {{x}^{2}}\cdot \sqrt[4]{{{y}^{3}}}\end{array}[/latex-display] The answer is [latex]\sqrt[4]{81x^{8}y^{3}}=3x^{2}\sqrt[4]{y^{3}} [/latex].

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. [latex] \sqrt[4]{81{{x}^{8}}{{y}^{3}}}[/latex]

Answer: Rewrite the radical using rational exponents. [latex-display] {{(81{{x}^{8}}{{y}^{3}})}^{\frac{1}{4}}}[/latex-display] Use the rules of exponents to simplify the expression. [latex-display] \begin{array}{r}{{81}^{\frac{1}{4}}}\cdot {{x}^{\frac{8}{4}}}\cdot {{y}^{\frac{3}{4}}}\\{{(3\cdot 3\cdot 3\cdot 3)}^{\frac{1}{4}}}{{x}^{2}}{{y}^{\frac{3}{4}}}\\{{({{3}^{4}})}^{\frac{1}{4}}}{{x}^{2}}{{y}^{\frac{3}{4}}}\\3{{x}^{2}}{{y}^{\frac{3}{4}}}\end{array}[/latex-display] Change the expression with the rational exponent back to radical form. [latex-display] 3{{x}^{2}}\sqrt[4]{{{y}^{3}}}[/latex-display]

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, [latex]\dfrac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}[/latex]. 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. [latex]\dfrac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}[/latex]

Answer: Separate the factors in the denominator. [latex-display] \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot \sqrt[3]{8}\cdot \sqrt[3]{{{b}^{4}}}}[/latex-display] Take the cube root of [latex]8[/latex], which is [latex]2[/latex]. [latex-display] \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot 2\cdot \sqrt[3]{{{b}^{4}}}}[/latex-display] Rewrite the radical using a fractional exponent. [latex-display] \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot 2\cdot {{b}^{\frac{4}{3}}}}[/latex-display] Rewrite the fraction as a series of factors in order to cancel factors (see next step). [latex-display] \frac{10}{2}\cdot \frac{{{c}^{2}}}{c}\cdot \frac{{{b}^{2}}}{{{b}^{\frac{4}{3}}}}[/latex-display] Simplify the constant and c factors. [latex-display] 5\cdot c\cdot \frac{{{b}^{2}}}{{{b}^{\frac{4}{3}}}}[/latex-display] Use the rule of negative exponents, n-x=[latex] \frac{1}{{{n}^{x}}}[/latex], to rewrite [latex] \frac{1}{{{b}^{\tfrac{4}{3}}}}[/latex] as [latex] {{b}^{-\tfrac{4}{3}}}[/latex]. [latex-display] 5c{{b}^{2}}{{b}^{-\ \frac{4}{3}}}[/latex-display] Combine the b factors by adding the exponents. [latex-display] 5c{{b}^{\frac{2}{3}}}[/latex-display] 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. [latex-display] 5c\sqrt[3]{{{b}^{2}}}[/latex-display]

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 [latex] \sqrt[n]{{{x}^{n}}}=x[/latex] if n is odd and [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex] 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, [latex] \sqrt[n]{{{x}^{n}}}=x[/latex].
  • If n is even, [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex]. (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 Learning License: 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.

CC licensed content, Shared previously

  • Precalculus. Provided by: OpenStax Authored by: Abramson, Jay. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download for free at : http://cnx.org/contents/[email protected]:1/Preface.
  • Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution.