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Study Guides > Intermediate Algebra

Simplify Expressions with Roots and Rational Exponents

11.1 Learning Objectives

  • Introduction to Roots
    • Define and evaluate principal square roots
    • Define and evaluate nth roots
    • Estimate roots that are not perfect
  • Radical Expressions and Rational Exponents
    • Define and identify a radical expression
    • Convert radicals to expressions with rational exponents
    • Convert expressions with rational exponents to their radical equivalent
  • Simplify Radical Expressions
    • Simplify radical expressions using factoring
    • Simplify radical expressions using rational exponents and the laws of exponents
    • Define x2=∣x∣\sqrt{x^2}=|x|, and apply it when simplifying radical expressions
Did you know that you can take the 6th root of a number? You have probably heard of a square root, written \sqrt{}, but you can also take a third, fourth and even a 5,000th root (if you really had to). In this lesson we will learn how a square root is defined and then we will build on that to form an understanding of nth roots.  We will use factoring and rules for exponents to simplify mathematical expressions that contain roots. The most common root is the square root. First, we will define what square roots are,  and how you find the square root of a number. Then we will apply similar ideas to define and evaluate nth roots. Roots are the inverse of exponents, much like multiplication is the inverse of division. Recall how exponents are defined, and written; with an exponent, as words, and as repeated multiplication. Exponent: 32 {{3}^{2}}, 45 {{4}^{5}}, x3 {{x}^{3}}, xn {{x}^{\text{n}}} Name: “Three squared” or “Three to the second power”, “Four to the fifth power”, “x cubed”, “x to the nth power” Repeated Multiplication: 3⋅3 3\cdot 3,  4⋅4⋅4⋅4⋅4 4\cdot 4\cdot 4\cdot 4\cdot 4,  x⋅x⋅x x\cdot x\cdot x,  x⋅x⋅x...⋅x⏟n times \underbrace{x\cdot x\cdot x...\cdot x}_{n\text{ times}}. Conversely,  when you are trying to find the square root of a number (say, 25), you are trying to find a number that can be multiplied by itself to create that original number. In the case of 25, you can find that 5⋅5=255\cdot5=25, so 5 must be the square root.

11.1.1 Square Roots

The symbol for the square root is called a radical symbol and looks like this:    \sqrt{\,\,\,}. The expression 25 \sqrt{25} is read “the square root of twenty-five” or “radical twenty-five.” The number that is written under the radical symbol is called the radicand. The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand. The following table shows different radicals and their equivalent written and simplified forms.
Radical Name Simplified Form
36 \sqrt{36} “Square root of thirty-six” “Radical thirty-six” 36=6⋅6=6 \sqrt{36}=\sqrt{6\cdot 6}=6
100 \sqrt{100} “Square root of one hundred” “Radical one hundred” 100=10⋅10=10 \sqrt{100}=\sqrt{10\cdot 10}=10
225 \sqrt{225} “Square root of two hundred twenty-five” “Radical two hundred twenty-five” 225=15⋅15=15 \sqrt{225}=\sqrt{15\cdot 15}=15
Consider 25 \sqrt{25} again. You may realize that there is another value that, when multiplied by itself, also results in 25. That number is −5−5.

5⋅5=25−5⋅−5=25 \begin{array}{r}5\cdot 5=25\\-5\cdot -5=25\end{array}

By definition, the square root symbol always means to find the positive root, called the principal root. So while 5⋅55\cdot5 and −5⋅−5−5\cdot−5 both equal 25, only 5 is the principal root. You should also know that zero is special because it has only one square root: itself (since 0⋅0=00\cdot0=0). In our first example we will show you how to use radical notation to evaluate principal square roots.

Example 11.1.a

Find the principal root of each expression.
  1. 100\sqrt{100}
  2. 16\sqrt{\sqrt{16}}
  3. 25+144\sqrt{25+144}
  4. 49−81\sqrt{49}-\sqrt{81}\\
  5. −81 -\sqrt{81}
  6. −9\sqrt{-9}

Answer:

  1. 100=10\sqrt{100}=10 because 102=100{10}^{2}=100
  2. 16=4=2\sqrt{\sqrt{16}}=\sqrt{4}=2 because 42=16{4}^{2}=16 and 22=4{2}^{2}=4
  3. Recall that square roots act as grouping symbols in the order of operations, so addition and subtraction must be performed first when they occur under a radical. 25+144=169=13\sqrt{25+144}=\sqrt{169}=13 because 132=169{13}^{2}=169
  4. This problem is similar to the last one, but this time subtraction should occur after evaluating the root. Stop and think about why these two problems are different. 49−81=7−9=−2\sqrt{49}-\sqrt{81}=7 - 9=-2 because 72=49{7}^{2}=49 and 92=81{9}^{2}=81

  5. The negative in front means to take the opposite of the value after you simplify the radical. −81−9⋅9 -\sqrt{81}\\-\sqrt{9\cdot 9}.  The square root of 81 is 9. Then, take the opposite of 9. −(9)−(9)

  6. −9\sqrt{-9}, we are looking for a number that when it is squared, returns −9-9. We can try (−3)2(-3)^2, but that will give a positive result, and 323^2 will also give a positive result. This leads to an important fact -  you cannot find the square root of a negative number.

In the following video we present more examples of how to find a principle square root. https://youtu.be/2cWAkmJoaDQ The last example we showed leads to an important characteristic of square roots. You can only take the square root of values that are nonnegative.

Domain of a Square Root −a\sqrt{-a} is not defined for all real numbers, a. Therefore, a\sqrt{a} is defined for a≥0a\ge0

Think About It

Does 25=±5\sqrt{25}=\pm 5? Write your ideas and a sentence to defend them in the box below before you look at the answer. [practice-area rows="1"][/practice-area]

Answer: No. Although both 52{5}^{2} and (−5)2{\left(-5\right)}^{2} are 2525, the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is 25=5\sqrt{25}=5.

11.1.2 Cube Roots

We know that 52=25, and 25=55^2=25, \text{ and }\sqrt{25}=5 but what if we want to "undo" 53=125, or 54=6255^3=125, \text{ or }5^4=625? We can use higher order roots to answer these questions. Suppose we know that a3=8{a}^{3}=8. We want to find what number raised to the 3rd power is equal to 8. Since 23=8{2}^{3}=8, we say that 2 is the cube root of 8. In the next example we will evaluate the cube roots of some perfect cubes.

Example 11.1.b

Evaluate the following:
  1. 1253 \sqrt[3]{125}
  2. −83 \sqrt[3]{-8}
  3. 273 \sqrt[3]{27}

Answer: 1. You can read this as “the third root of 125” or “the cube root of 125.” To evaluate this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. ?⋅?⋅?=125\text{?}\cdot\text{?}\cdot\text{?}=125. Since 125 ends in 5, 5 is a good candidate. 5⋅5⋅5=1255\cdot5\cdot5=125 2. We want to find a number whose cube is -8. We know 2 is the cube root of 8, so maybe we can try -2. Since −2⋅−2⋅−2=−8-2\cdot{-2}\cdot{-2}=-8, the cube root of -8 is -2. This is different from square roots because multiplying three negative numbers together results in a negative number. 3.  We want to find a number whose cube is 27. Since 3⋅3⋅3=273\cdot3\cdot3=27, the cube root of 27 is 3.

As we saw in the last example,there is one interesting fact about cube roots that is not true of square roots. Negative numbers can’t have real number square roots, but negative numbers can have real number cube roots! What is the cube root of −8−8? −83=−2 \sqrt[3]{-8}=-2 because −2⋅−2⋅−2=−8 -2\cdot -2\cdot -2=-8. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider (−1)33=−1 \sqrt[3]{{{(-1)}^{3}}}=-1. In the following video we show more examples of finding a cube root. https://youtu.be/9Nh-Ggd2VJo

11.1.3 Nth Roots

The cube root of a number is written with a small number 3, called the index, just outside and above the radical symbol. It looks like 3 \sqrt[3]{{}}. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol. We can apply the same idea to any exponent and it's corresponding root.  The nth root of aa is a number that, when raised to the nth power, gives aa. For example, 33 is the 5th root of 243243 because (3)5=243{\left(3\right)}^{5}=243. If aa is a real number with at least one nth root, then the principal nth root of aa is the number with the same sign as aa that, when raised to the nth power, equals aa. The principal nth root of aa is written as an\sqrt[n]{a}, where nn is a positive integer greater than or equal to 2. In the radical expression, nn is called the index of the radical.

Definition: Principal nth Root

If aa is a real number with at least one nth root, then the principal nth root of aa, written as an\sqrt[n]{a}, is the number with the same sign as aa that, when raised to the nth power, equals aa. The index of the radical is nn.

Example 11.1.c

Evaluate each of the following:
  1. −325\sqrt[5]{-32}
  2. 814\sqrt[4]{81}
  3. −18\sqrt[8]{-1}

Answer:

  1. −325\sqrt[5]{-32}. Factor -32. We know that −2⋅−2⋅−2⋅−2⋅−2=−32-2\cdot-2\cdot-2\cdot-2\cdot-2=-32 which means (−2)5=−32{\left(-2\right)}^{5}=-32. Therefore,−325\sqrt[5]{-32}=(−2)55=−2\sqrt[5]{\left(-2\right)^5}=-2
  2. 814\sqrt[4]{81}. Factoring can help.  We know that 9⋅9=819\cdot9=81 and we can further factor each 9: 814=3⋅3⋅3⋅34=344=3\sqrt[4]{81}=\sqrt[4]{3\cdot3\cdot3\cdot3}=\sqrt[4]{3^4}=3
  3. −18\sqrt[8]{-1}, since we have an 8th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, −1⋅−1⋅−1⋅−1⋅−1⋅−1⋅−1⋅−1=+1-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1=+1

In the following video we show more examples of how to evaluate and nth root. https://youtu.be/vA2DkcUSRSk You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can evaluate the radicals −813, −645 \sqrt[3]{-81},\ \sqrt[5]{-64}, and −21877 \sqrt[7]{-2187}, but you cannot evaluate the radicals −100, −164 \sqrt[{}]{-100},\ \sqrt[4]{-16}, or −2,5006 \sqrt[6]{-2,500}.

11.1.4 Estimate Roots

An approach to handling roots that are not perfect (squares, cubes, etc.)  is to approximate them by comparing the values to perfect squares, cubes, or nth roots. Suppose you wanted to know the square root of 17. Let’s look at how you might approximate it.

Example 11.1.d

Estimate. 17 \sqrt{17}

Answer: Think of two perfect squares that surround 17. 17 is in between the perfect squares 16 and 25. So, 17 \sqrt{17} must be in between 16 \sqrt{16} and 25 \sqrt{25}. Determine whether 17 \sqrt{17} is closer to 4 or to 5 and make another estimate.

16=4 \sqrt{16}=4 and 25=5 \sqrt{25}=5

Since 17 is closer to 16 than 25, 17 \sqrt{17} is probably about 4.1 or 4.2. Use trial and error to get a better estimate of 17 \sqrt{17}. Try squaring incrementally greater numbers, beginning with 4.1, to find a good approximation for 17 \sqrt{17}.

(4.1)2\left(4.1\right)^{2}

(4.1)2\left(4.1\right)^{2} gives a closer estimate than (4.2)2(4.2)^{2}.

4.1⋅4.1=16.814.2⋅4.2=17.644.1\cdot4.1=16.81\\4.2\cdot4.2=17.64

Continue to use trial and error to get an even better estimate.

4.12⋅4.12=16.97444.13⋅4.13=17.05694.12\cdot4.12=16.9744\\4.13\cdot4.13=17.0569

Answer

17≈4.12 \sqrt{17}\approx 4.12

This approximation is pretty close. If you kept using this trial and error strategy you could continue to find the square root to the thousandths, ten-thousandths, and hundred-thousandths places, but eventually it would become too tedious to do by hand. For this reason, when you need to find a more precise approximation of a square root, you should use a calculator. Most calculators have a square root key () (\sqrt{{}}) that will give you the square root approximation quickly. On a simple 4-function calculator, you would likely key in the number that you want to take the square root of and then press the square root key. Try to find 17 \sqrt{17} using your calculator. Note that you will not be able to get an “exact” answer because 17 \sqrt{17} is an irrational number, a number that cannot be expressed as a fraction, and the decimal never terminates or repeats. To nine decimal positions, 17 \sqrt{17} is approximated as 4.123105626. A calculator can save a lot of time and yield a more precise square root when you are dealing with numbers that aren’t perfect squares.

Example 11.1.e

Approximate 303 \sqrt[3]{30} and also find its value using a calculator.

Answer: Find the cubes that surround 30. 30 is in between the perfect cubes 27 and 64. 273=3 \sqrt[3]{27}=3 and 643=4 \sqrt[3]{64}=4, so 303 \sqrt[3]{30} is between 3 and 4. Use a calculator.

303≈3.10723\sqrt[3]{30}\approx3.10723

Answer

By approximation: 3≤303≤43\le\sqrt[3]{30}\le4 Using a calculator: 303≈3.10723 \sqrt[3]{30}\approx3.10723

The following video shows another example of how to estimate a square root. https://youtu.be/iNfalyW7olk

11.1.5 Radical Expressions and Rational Exponents

Square roots are most often written using a radical sign, like this, 4 \sqrt{4}. But there is another way to represent them. You can use rational exponents instead of a radical. A rational exponent is an exponent that is a fraction. For example, 4 \sqrt{4} can be written as 412 {{4}^{\tfrac{1}{2}}}. Can’t imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems. Writing radicals with rational exponents will come in handy when we discuss techniques for simplifying more complex radical expressions. Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as16 \sqrt{16}, to quite complicated, as in 250x4y3 \sqrt[3]{250{{x}^{4}}y}

11.1.6 Write an expression with a rational exponent as a radical

Radicals and fractional exponents are alternate ways of expressing the same thing.  In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.

Radical Form

Exponent Form

Principal Root
16 \sqrt{16} 1612 {{16}^{\tfrac{1}{2}}} 4
25 \sqrt{25} 2512 {{25}^{\tfrac{1}{2}}} 5
100 \sqrt{100} 10012 {{100}^{\tfrac{1}{2}}} 10
Let’s look at some more examples, but this time with cube roots. Remember, cubing a number raises it to the power of three. Notice that in the examples in the table below, the denominator of the rational exponent is the number 3.

Radical Form

Exponent Form

Principal Root
83 \sqrt[3]{8} 813 {{8}^{\tfrac{1}{3}}} 2
83 \sqrt[3]{8} 12513 {{125}^{\tfrac{1}{3}}} 5
10003 \sqrt[3]{1000} 100013 {{1000}^{\tfrac{1}{3}}} 10
These examples help us model a relationship between radicals and rational exponents: namely, that the nth root of a number can be written as either xn \sqrt[n]{x} or x1n {{x}^{\frac{1}{n}}}.

Radical Form

Exponent Form
x \sqrt{x} x12 {{x}^{\tfrac{1}{2}}}
x3 \sqrt[3]{x} x13 {{x}^{\tfrac{1}{3}}}
x4 \sqrt[4]{x} x14 {{x}^{\tfrac{1}{4}}}
… …
xn \sqrt[n]{x} x1n {{x}^{\tfrac{1}{n}}}
In the table above, notice how the denominator of the rational exponent determines the index of the root. So, an exponent of 12 \frac{1}{2} translates to the square root, an exponent of 15 \frac{1}{5} translates to the fifth root or 5 \sqrt[5]{{}}, and 18 \frac{1}{8} translates to the eighth root or 8 \sqrt[8]{{}}.

Example 11.1.f

Express (2x)13 {{(2x)}^{^{\frac{1}{3}}}} in radical form.

Answer: Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.

2x3\sqrt[3]{2x}

The parentheses in (2x)13 {{\left( 2x \right)}^{\frac{1}{3}}} indicate that the exponent refers to everything within the parentheses.

Answer

(2x)13=2x3 {{(2x)}^{^{\frac{1}{3}}}}=\sqrt[3]{2x}

Remember that exponents only refer to the quantity immediately to their left unless a grouping symbol is used. The example below looks very similar to the previous example with one important difference—there are no parentheses! Look what happens.

Example 11.1.g

Express 2x13 2{{x}^{^{\frac{1}{3}}}} in radical form.

Answer: Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.

2x3 2\sqrt[3]{x}

The exponent refers only to the part of the expression immediately to the left of the exponent, in this case x, but not the 2.

Answer

2x13=2x3 2{{x}^{^{\frac{1}{3}}}}=2\sqrt[3]{x}

11.1.7 Write a radical expression as an expression with a rational exponent

Person sitting on the ground with one leg arched behind them and one leg curved in front of them. Flexibility
We can write radicals with rational exponents, and as we will see when we simplify more complex radical expressions, this can make things easier. Having different ways to express and write algebraic expressions allows us to have flexibility in solving and simplifying them. It is like having a thesaurus when you write, you want to have options for expressing yourself!

Example 11.1.h

Write 814 \sqrt[4]{81} as an expression with a rational exponent.

Answer: The radical form     4 \Large\sqrt[4]{{\,\,\,\,}} can be rewritten as the exponent 14 \frac{1}{4}. Remove the radical and place the exponent next to the base. 8114 {{81}^{\frac{1}{4}}}

Answer

814=8114 \sqrt[4]{81}={{81}^{\frac{1}{4}}}

Example 11.1.i

Express 4xy3 4\sqrt[3]{xy} with rational exponents.

Answer: Rewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is 3, so the rational exponent will be 13 \frac{1}{3}.

4(xy)13 4{{(xy)}^{\frac{1}{3}}}

Since 4 is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.

Answer

4xy3=4(xy)13 4\sqrt[3]{xy}=4{{(xy)}^{\frac{1}{3}}}

11.1.8 Rational exponents whose numerator is not equal to one

All of the numerators for the fractional exponents in the examples above were 1. You can use fractional exponents that have numerators other than 1 to express roots, as shown below.

Radical

Exponent
9 \sqrt{9} 9129^{\frac{1}{2}}
923 \sqrt[3]{{{9}^{2}}} 9239^{\frac{2}{3}}
934\sqrt[4]{9^{3}} 9349^{\frac{3}{4}}
925\sqrt[5]{9^{2}} 9259^{\frac{2}{5}}
… …
9xn\sqrt[n]{9^{x}} 9xn9\frac{x}{n}
Screen Shot 2016-07-29 at 3.56.45 PM To rewrite a radical using a fractional exponent, the power to which the radicand is raised becomes the numerator and the root/ index becomes the denominator.

Writing Rational Exponents

Any radical in the form axn\sqrt[n]{a^{x}}  can be written using a fractional exponent in the form axna^{\frac{x}{n}}.
The relationship between axn \sqrt[n]{{{a}^{x}}}and axn {{a}^{\frac{x}{n}}} works for rational exponents that have a numerator of 1 as well. For example, the radical 83 \sqrt[3]{8} can also be written as 813 \sqrt[3]{{{8}^{1}}}, since any number remains the same value if it is raised to the first power. You can now see where the numerator of 1 comes from in the equivalent form of 813 {{8}^{\frac{1}{3}}}. In the next example, we practice writing radicals with rational exponents where the numerator is not equal to one.

Example 11.1.j

Rewrite the radicals using a rational exponent, then simplify your result.
  1. a63\sqrt[3]{{{a}^{6}}}
  2. 16312\sqrt[12]{16^3}

Answer: 1.axn\sqrt[n]{a^{x}} can be rewritten as axna^{\frac{x}{n}}, so in this case n=3, and x=6n=3,\text{ and }x=6.  Therefore a63=a63\sqrt[3]{{{a}^{6}}}={{a}^{\frac{6}{3}}} Simplify the exponent. a63=a2{{a}^{\frac{6}{3}}}={{a}^{2}}

Answer

a63=a2 \sqrt[3]{{{a}^{6}}}={{a}^{2}} 2. axn\sqrt[n]{a^{x}} can be rewritten as axna^{\frac{x}{n}}, so in this case a=16,n=12, and x=3a=16, n=12,\text{ and }x=3.  Therefore

16312=16312=1614\sqrt[12]{16^3}={16}^{\frac{3}{12}}={16}^{\frac{1}{4}}

Simplify the expression using rules for exponents.

16=241614=2414=24⋅14=21=2\begin{array}{ccc}16=2^4\\{16}^{\frac{1}{4}}={2^4}^{\frac{1}{4}}\\=2^{4\cdot\frac{1}{4}}\\=2^1=2\end{array}

Answer

16312=2\sqrt[12]{16^3}=2

In our last example we will rewrite expressions with rational exponents as radicals. This practice will help us when we simplify more complicated radical expressions, and as we learn how to solve radical equations. Typically it is easier to simplify when we use rational exponents, but this exercise is intended to help you understand how the numerator and denominator of the exponent are the exponent of a radicand and index of a radical.

Example 11.1.k

Rewrite the expressions using a radical.
  1. x23{x}^{\frac{2}{3}}
  2. 547{5}^{\frac{4}{7}}

Answer:

  1. x23{x}^{\frac{2}{3}}, the numerator is 2 and the denominator is 3, therefore we will have the third root of x squared, x23\sqrt[3]{x^2}
  2. 547{5}^{\frac{4}{7}}, the numerator is 4 and the denominator is 7, so we will have the seventh root of 5 raised to the fourth power. 547\sqrt[7]{5^4}

In the following video we show more examples of writing radical expressions with rational exponents and expressions with rational exponents as radical expressions. https://youtu.be/5cWkVrANBWA We will use this notation later, so come back for practice if you forget how to write a radical with a rational exponent.

11.1.9 Simplify Radical Expressions

Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as16 \sqrt{16}, to quite complicated, as in 250x4y3 \sqrt[3]{250{{x}^{4}}y}. 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\left(ab\right)^{x}=a^{x}\cdot{b}^{x}. So, for example, you can use the rule to rewrite (3x)2 {{\left( 3x \right)}^{2}} as 32⋅x2=9⋅x2=9x2 {{3}^{2}}\cdot {{x}^{2}}=9\cdot {{x}^{2}}=9{{x}^{2}}. Now instead of using the exponent 2, let’s use the exponent 12 \frac{1}{2}. The exponent is distributed in the same way.

(3x)12=312⋅x12 {{\left( 3x \right)}^{\frac{1}{2}}}={{3}^{\frac{1}{2}}}\cdot {{x}^{\frac{1}{2}}}

And since you know that raising a number to the 12 \frac{1}{2} power is the same as taking the square root of that number, you can also write it this way.

3x=3⋅x \sqrt{3x}=\sqrt{3}\cdot \sqrt{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 \sqrt{ab}=\sqrt{a}\cdot \sqrt{b}. For example: 100=10⋅10 \sqrt{100}=\sqrt{10}\cdot \sqrt{10}, and 75=25⋅3 \sqrt{75}=\sqrt{25}\cdot \sqrt{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) \sqrt{(2\cdot 2)(2\cdot 2)(3\cdot 3}), you can rewrite the expression as the product of multiple perfect squares: 22⋅22⋅32 \sqrt{{{2}^{2}}}\cdot \sqrt{{{2}^{2}}}\cdot \sqrt{{{3}^{2}}}.

The square root of a product rule will help us simplify roots that aren't perfect, as is shown the following example.

Example 11.1.l

Simplify. 63 \sqrt{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⋅9 \sqrt{7\cdot 9} 9 is a perfect square, 9=329=3^2, therefore we can rewrite the radicand. 7⋅32 \sqrt{7\cdot {{3}^{2}}} Using the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical. 7⋅32 \sqrt{7}\cdot \sqrt{{{3}^{2}}} Take the square root of 323^{2}. 7⋅3 \sqrt{7}\cdot 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 3\cdot \sqrt{7} Answer 63=37 \sqrt{63}=3\sqrt{7}

The final answer 37 3\sqrt{7} 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 \sqrt{{{x}^{2}}}. 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 \sqrt{{{x}^{2}}}. Where are they equal? Where are they not equal? After doing that for each row, look again and determine whether the value of x2 \sqrt{{{x}^{2}}} is the same as the value of ∣x∣\left|x\right|.
xx x2x^{2} x2\sqrt{x^{2}} ∣x∣\left|x\right|
−5−5 25 5 5
−2−2 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\sqrt{x^{2}}\neq{x}! However, in all cases x2=∣x∣\sqrt{x^{2}}=\left|x\right|. You need to consider this fact when simplifying radicals with an even index that contain variables, because by definition x2\sqrt{x^{2}} 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∣\sqrt{x^{2}}=\left|x\right|. Examples: 9x2=3∣x∣\sqrt{9x^{2}}=3\left|x\right|, and 16x2y2=4∣xy∣\sqrt{16{{x}^{2}}{{y}^{2}}}=4\left|xy\right|
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 11.1.m

Simplify. a3b5c2 \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}

Answer: Factor to find variables with even exponents. a2⋅a⋅b4⋅b⋅c2 \sqrt{{{a}^{2}}\cdot a\cdot {{b}^{4}}\cdot b\cdot {{c}^{2}}} Rewrite b4b^{4} as (b2)2\left(b^{2}\right)^{2}. a2⋅a⋅(b2)2⋅b⋅c2 \sqrt{{{a}^{2}}\cdot a\cdot {{({{b}^{2}})}^{2}}\cdot b\cdot {{c}^{2}}} Separate the squared factors into individual radicals. a2⋅(b2)2⋅c2⋅a⋅b \sqrt{{{a}^{2}}}\cdot \sqrt{{{({{b}^{2}})}^{2}}}\cdot \sqrt{{{c}^{2}}}\cdot \sqrt{a\cdot b} Take the square root of each radical. Remember that a2=∣a∣ \sqrt{{{a}^{2}}}=\left| a \right|. ∣a∣⋅b2⋅∣c∣⋅a⋅b \left| a \right|\cdot {{b}^{2}}\cdot \left|{c}\right|\cdot \sqrt{a\cdot b} Simplify and multiply. ∣ac∣b2ab \left| ac \right|{{b}^{2}}\sqrt{ab}

Answer

a3b5c2=∣ac∣b2ab \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\left| ac \right|{{b}^{2}}\sqrt{ab}

Analysis of the Solution

Why didn't we write b2b^2 as ∣b2∣|b^2|?  Because when you square a number, you will always get a positive result, so the principal square root of (b2)2\left(b^2\right)^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 11.1.n

Simplify. 9x6y4 \sqrt{9{{x}^{6}}{{y}^{4}}}

Answer: Factor to find identical pairs.

3⋅3⋅x3⋅x3⋅y2⋅y2 \sqrt{3\cdot 3\cdot {{x}^{3}}\cdot {{x}^{3}}\cdot {{y}^{2}}\cdot {{y}^{2}}}

Rewrite the pairs as perfect squares.

32⋅(x3)2⋅(y2)2 \sqrt{{{3}^{2}}\cdot {{\left( {{x}^{3}} \right)}^{2}}\cdot {{\left( {{y}^{2}} \right)}^{2}}}

Separate into individual radicals.

32⋅(x3)2⋅(y2)2 \sqrt{{{3}^{2}}}\cdot \sqrt{{{\left( {{x}^{3}} \right)}^{2}}}\cdot \sqrt{{{\left( {{y}^{2}} \right)}^{2}}}

Simplify.

3x3y2 3{{x}^{3}}{{y}^{2}}

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

3∣x3∣y2 3|{{x}^{3}}|{{y}^{2}}

Answer

9x6y4=3∣x3∣y \sqrt{9{{x}^{6}}{{y}^{4}}}=3|{{x}^{3}}|{y}

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 11.1.o

Simplify. (36x4)12 {{(36{{x}^{4}})}^{\frac{1}{2}}}

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

36x4 \sqrt{36{{x}^{4}}}

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

62⋅x462⋅x462⋅(x2)26⋅x2\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}

Answer

(36x4)12=6x2 {{(36{{x}^{4}})}^{\frac{1}{2}}}=6{{x}^{2}}

Here is one more example with perfect squares.

Example 11.1.p

Simplify. 49x10y8 \sqrt{49{{x}^{10}}{{y}^{8}}}

Answer: Look for squared numbers and variables. Factor 49 into 7⋅77\cdot7, x10x^{10} into x5⋅x5x^{5}\cdot{x}^{5}, and y8y^{8} into y4⋅y4y^{4}\cdot{y}^{4}.

7⋅7⋅x5⋅x5⋅y4⋅y4 \sqrt{7\cdot 7\cdot {{x}^{5}}\cdot {{x}^{5}}\cdot {{y}^{4}}\cdot {{y}^{4}}}

Rewrite the pairs as squares.

72⋅(x5)2⋅(y4)2 \sqrt{{{7}^{2}}\cdot {{({{x}^{5}})}^{2}}\cdot {{({{y}^{4}})}^{2}}}

Separate the squared factors into individual radicals.

72⋅(x5)2⋅(y4)2 \sqrt{{{7}^{2}}}\cdot \sqrt{{{({{x}^{5}})}^{2}}}\cdot \sqrt{{{({{y}^{4}})}^{2}}}

Take the square root of each radical using the rule that x2=x \sqrt{{{x}^{2}}}=x.

7⋅x5⋅y4 7\cdot {{x}^{5}}\cdot {{y}^{4}}

Multiply.

7x5y4 7{{x}^{5}}{{y}^{4}}

Answer

49x10y8=7∣x5∣y4 \sqrt{49{{x}^{10}}{{y}^{8}}}=7|{{x}^{5}}|{{y}^{4}}

11.1.10 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 11.1.q

Simplify. 40m53 \sqrt[3]{40{{m}^{5}}}

Answer: Factor 40 into prime factors. 5⋅2⋅2⋅2⋅m53 \sqrt[3]{5\cdot 2\cdot 2\cdot 2\cdot {{m}^{5}}} Since you are looking for the cube root, you need to find factors that appear 3 times under the radical. Rewrite 2⋅2⋅2 2\cdot 2\cdot 2 as 23 {{2}^{3}}. 23⋅5⋅m53 \sqrt[3]{{{2}^{3}}\cdot 5\cdot {{m}^{5}}} Rewrite m5 {{m}^{5}} as m3⋅m2 {{m}^{3}}\cdot {{m}^{2}}. 23⋅5⋅m3⋅m23 \sqrt[3]{{{2}^{3}}\cdot 5\cdot {{m}^{3}}\cdot {{m}^{2}}} Rewrite the expression as a product of multiple radicals. 233⋅53⋅m33⋅m23 \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{5}\cdot \sqrt[3]{{{m}^{3}}}\cdot \sqrt[3]{{{m}^{2}}} Simplify and multiply. 2⋅53⋅m⋅m23 2\cdot \sqrt[3]{5}\cdot m\cdot \sqrt[3]{{{m}^{2}}}

Answer

40m53=2m5m23 \sqrt[3]{40{{m}^{5}}}=2m\sqrt[3]{5{{m}^{2}}}

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 11.1.r

Simplify. −27x4y33 \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}

Answer: Factor the expression into cubes. Separate the cubed factors into individual radicals. −1⋅27⋅x4⋅y33(−1)3⋅(3)3⋅x3⋅x⋅y33(−1)33⋅(3)33⋅x33⋅x3⋅y33\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} Simplify the cube roots. −1⋅3⋅x⋅y⋅x3 -1\cdot 3\cdot x\cdot y\cdot \sqrt[3]{x}

Answer

−27x4y33=−3xyx3 \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\sqrt[3]{x}

You could check your answer by performing the inverse operation. If you are right, when you cube −3xyx3 -3xy\sqrt[3]{x} you should get −27x4y3 -27{{x}^{4}}{{y}^{3}}. (−3xyx3)(−3xyx3)(−3xyx3)−3⋅−3⋅−3⋅x⋅x⋅x⋅y⋅y⋅y⋅x3⋅x3⋅x3−27⋅x3⋅y3⋅x33−27x3y3⋅x−27x4y3 \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} You can also skip the step of factoring out the negative one once you are comfortable with identifying cubes.

Example 11.1.s

Simplify. −24a53 \sqrt[3]{-24{{a}^{5}}}

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

−1⋅8⋅3⋅a53 \sqrt[3]{-1\cdot 8\cdot 3\cdot {{a}^{5}}}

Factor variables. You are looking for cube exponents, so you factor a5a^{5} into a3a^{3} and a2a^{2}.

(−1)3⋅23⋅3⋅a3⋅a23 \sqrt[3]{{{(-1)}^{3}}\cdot {{2}^{3}}\cdot 3\cdot {{a}^{3}}\cdot {{a}^{2}}}

Separate the factors into individual radicals.

(−1)33⋅233⋅a33⋅3⋅a23 \sqrt[3]{{{(-1)}^{3}}}\cdot \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{{{a}^{3}}}\cdot \sqrt[3]{3\cdot {{a}^{2}}}

Simplify, using the property x33=x \sqrt[3]{{{x}^{3}}}=x. 

−1⋅2⋅a⋅3⋅a23 -1\cdot 2\cdot a\cdot \sqrt[3]{3\cdot {{a}^{2}}}

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

−2a3a23 -2a\sqrt[3]{3{{a}^{2}}}

Answer

−24a53=−2a3a23 \sqrt[3]{-24{{a}^{5}}}=-2a\sqrt[3]{3{{a}^{2}}}

In the following video we show more examples of simplifying cube roots. https://youtu.be/BtJruOpmHCE

11.1.11 Simplifying fourth roots

Now let's 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 11.1.t

Simplify. 81x8y34 \sqrt[4]{81{{x}^{8}}{{y}^{3}}}

Answer: Rewrite the expression. 814⋅x84⋅y34 \sqrt[4]{81}\cdot \sqrt[4]{{{x}^{8}}}\cdot \sqrt[4]{{{y}^{3}}} Factor each radicand. 3⋅3⋅3⋅34⋅x2⋅x2⋅x2⋅x24⋅y34 \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}}} Simplify. 344⋅(x2)44⋅y343⋅x2⋅y34\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}

Answer

81x8y34=3x2y34\sqrt[4]{81x^{8}y^{3}}=3x^{2}\sqrt[4]{y^{3}}

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 11.1.u

Simplify. 81x8y34 \sqrt[4]{81{{x}^{8}}{{y}^{3}}}

Answer: Rewrite the radical using rational exponents. (81x8y3)14 {{(81{{x}^{8}}{{y}^{3}})}^{\frac{1}{4}}} Use the rules of exponents to simplify the expression. 8114⋅x84⋅y34(3⋅3⋅3⋅3)14x2y34(34)14x2y343x2y34 \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} Change the expression with the rational exponent back to radical form. 3x2y34 3{{x}^{2}}\sqrt[4]{{{y}^{3}}}

Answer

81x8y34=3x2y34 \sqrt[4]{81{{x}^{8}}{{y}^{3}}}=3{{x}^{2}}\sqrt[4]{{{y}^{3}}}

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, 10b2c2c8b43\large\frac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}. This expression has two variables, a fraction, and a radical. Let’s 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 or a fraction can be rewritten as a negative exponent.

Example 11.1.v

Simplify. 10b2c2c8b43\large\frac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}

Answer: Separate the factors in the denominator. 10b2c2c⋅83⋅b43 \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot \sqrt[3]{8}\cdot \sqrt[3]{{{b}^{4}}}} Take the cube root of 8, which is 2. 10b2c2c⋅2⋅b43 \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot 2\cdot \sqrt[3]{{{b}^{4}}}} Rewrite the radical using a fractional exponent. 10b2c2c⋅2⋅b43 \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot 2\cdot {{b}^{\frac{4}{3}}}} Rewrite the fraction as a series of factors in order to cancel factors (see next step). 102⋅c2c⋅b2b43 \frac{10}{2}\cdot \frac{{{c}^{2}}}{c}\cdot \frac{{{b}^{2}}}{{{b}^{\frac{4}{3}}}} Simplify the constant and c factors. 5⋅c⋅b2b43 5\cdot c\cdot \frac{{{b}^{2}}}{{{b}^{\frac{4}{3}}}} Use the rule of negative exponents, n-x=1nx \frac{1}{{{n}^{x}}}, to rewrite 1b43 \frac{1}{{{b}^{\tfrac{4}{3}}}} as b−43 {{b}^{-\tfrac{4}{3}}}. 5cb2b− 43 5c{{b}^{2}}{{b}^{-\ \frac{4}{3}}} Combine the b factors by adding the exponents. 5cb23 5c{{b}^{\frac{2}{3}}} 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. 5cb23 5c\sqrt[3]{{{b}^{2}}}

Answer

10b2c2c8b43=5cb23 \frac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}=5c\sqrt[3]{{{b}^{2}}}

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 xnn=x \sqrt[n]{{{x}^{n}}}=x if n is odd, and xnn=∣x∣ \sqrt[n]{{{x}^{n}}}=\left| x \right| 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, xnn=x \sqrt[n]{{{x}^{n}}}=x.
  • If n is even, xnn=∣x∣ \sqrt[n]{{{x}^{n}}}=\left| x \right|. (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.)

Summary

The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are greater than or equal to 0. The square root of a perfect square will be an integer. Other roots can be simplified by identifying factors that are perfect squares, cubes, etc. Nth roots can be approximated using trial and error or a calculator. Any radical in the form axn\sqrt[n]{a^{x}}  can be written using a fractional exponent in the form axna^{\frac{x}{n}}. Rewriting radicals using fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.

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