Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

10.1: Simplify Radicals

( \newcommand{\kernel}{\mathrm{null}\,}\)

Not all radicands are perfect squares, where when we take the square root, we obtain a positive integer. For example, if we input 8 in a calculator, the calculator would display 2.828427124746190097603377448419 and even this number is a rounded approximation of the square root. To be as accurate as possible, we will leave all answers in exact form, i.e., answers contain integers and radicals- no decimals.

Note

When we say to simplify an expression with radicals, the simplified expression should have

  • a radical, unless the radical reduces to an integer
  • a radicand with no factors containing perfect squares
  • no decimals

Following these guidelines ensures the expression is in its simplest form.

Simplify Radicals

Product Rule for Radicals

If a, b are any two positive real numbers, then ab=ab

In general, if a, b are any two positive real numbers, then nab=nanb,
where n is a positive integer and n2.

Example 10.1.1

Simplify: 75

Solution

We can apply the product rule for radicals to simplify this number. We need to find the largest factor of 75 that is a perfect square (since we have a square root) and rewrite the radicand as a product of this perfect square and its other factor. The largest factor of radicand 75 that is a perfect square is 25.

75Rewrite radicand as a product of 25 and 3253Apply product rule for radicals253Simplify each square root53Rewrite53Simplified expression

If the radicand is not a perfect square, we leave as is; hence, we left 3 as is.

Example 10.1.2

Simplify: 72

Solution

We can apply the product rule for radicals to simplify this number. We need to find the largest factor of 72 that is a perfect square (since we have a square root) and rewrite the radicand as a product of this perfect square and its other factor. The largest factor of radicand 72 that is a perfect square is 36.

72Rewrite radicand as a product of 36 and 2362Apply product rule for radicals362Simplify each square root62Rewrite62Simplified expression

If the radicand is not a perfect square, we leave as is; hence, we left 2 as is.

Simplify Radicals with Coefficients

Example 10.1.3

Simplify: 563

Solution

We can apply the product rule for radicals to simplify this number and multiply coefficients in the last steps. We need to find the largest factor of 63 that is a perfect square (since we have a square root) and rewrite the radicand as a product of this perfect square and its other factor. The largest factor of radicand 63 that is a perfect square is 9.

563Rewrite radicand as a product of 9 and 7597Apply product rule for radicals597Simplify each square root537Rewrite and simplify coefficients157Simplified expression

If the radicand is not a perfect square, we leave as is; hence, we left 7 as is.

Rational Exponents

When we simplify radicals, we extract roots of factors with exponents in which are multiples of the root (index). For example, x4=2x4=x2, but notice we just divided the power on x by the root. Let’s look at the example again, but now as division of exponents:

x4=3x4=x42=x2

Division with exponents, or fraction exponents, are called rational exponents.

Definition: Rational Exponent

Let a be the base, and m and n be real real numbers. Then

amn=nam=(na)m

The denominator of a rational exponent is the root on the radical and vice versa

Example 10.1.4

Rewrite each radical with its corresponding rational exponent.

  1. (5x)3
  2. (63x)5
  3. 1(7a)3
  4. 1(3xy)2

Solution

  1. For the expression (5x)3, we see the root is 5. This means that the denominator of the rational exponent is 5. Hence, the numerator is the exponent 3: (5x)3=x35.
  2. For the expression (63x)5, we see the root is 6. This means that the denominator of the rational exponent is 6. Hence, the numerator is the exponent 5: (63x)5=(3x)56.
  3. For the expression 1(7a)3, we see the root is 7. This means that the denominator of the rational exponent is 7. Hence, the numerator is the exponent 3. Furthermore, since the expression with the radical is in the denominator, we can rewrite the expression using a negative exponent: 1(7a)3=(a)37.
  4. For the expression 1(3xy)2, we see the root is 3. This means that the denominator of the rational exponent is 3. Hence, the numerator is the exponent 2. Furthermore, since the expression with the radical is in the denominator, we can rewrite the expression using a negative exponent: 1(3xy)2=(xy)23.
Example 10.1.5

Rewrite each expression in its equivalent radical form.

  1. a53
  2. (2mn)27
  3. x45
  4. (xy)29

Solution

  1. From the definition, we know that the denominator of the rational exponent is the root making the numerator the power: a53=3a5 or (3a)5.
  2. From the definition, we know that the denominator of the rational exponent is the root making the numerator the power: (2mn)27=7(2mn)2 or (72mn)2.
  3. From the definition, we know that the denominator of the rational exponent is the root making the numerator the power: x45=(5x)4. Notice that the expression still contains a negative exponent. Hence, we need to reciprocate the radical to rewrite the expression with only positive exponents: x45=1(5x)4
  4. From the definition, we know that the denominator of the rational exponent is the root making the numerator the power: (xy)29=(9x)2. Notice that the expression still contains a negative exponent. Hence, we need to reciprocate the radical to rewrite the expression with only positive exponents: (xy)29=1(9xy)2
Note

Nicole Oresme, a Mathematician born in Normandy was the first to use rational exponents. He used the notation 139p to represent 913. However, his notation went largely unnoticed

The ability to change between rational exponential expressions and radical expressions allows us to evaluate expressions.

Example 10.1.6

Evaluate 2743.

Solution

We first rewrite the expression with only positive exponents, then evaluate the exponen

2743Rewrite the expression with positive exponents12743Rewrite in radical form1(327)4Evaluate radical 327=31(3)4Evaluate exponent 34=81181Result

Thus, 2743=181. This result should emphasize the fact that negative exponents means reciprocals, and not negative numbers.

Simplify Radicals with Variables

Commonly, radicands can contain variables. When taking the square roots of variables, we know the root is 2; we do not always write it, but we know it’s there. Hence, we apply the product rule of radicals by rewriting the variable’s exponent and rewrite the exponents so that one of the exponents is the largest even number.

Example 10.1.7

Simplify: x6y5

Solution

We can apply the product rule for radicals to simplify by rewriting the variable’s exponent and rewrite the exponents so that one of the exponents is the largest even number.

x6y5Rewrite radicandx6y4y1Apply product rule for radicalsx6y4ySimplify each square rootx3y2yRewrite and simplify coefficientsx3y2ySimplified expression

Notice that (x3)2 and (y2)2=y4; hence, we extract the perfect squares of the variables and leave the y as is.

Note

Recall, when taking a square root of a number, the radicand must be greater than or equal to zero. So, when we are applying the square root to variables, the variables must also be greater than or equal to zero.

Notice, we are essentially dividing the exponents on the variables by two and the factor that remains in the radicand has exponent 1.

Example 10.1.8

Simplify: 518x4y6z10. Assume all variables are positive.

Solution

We can apply the product rule for radicals to simplify by rewriting the variable’s exponent and rewrite the exponents so that one of the exponents is the largest even number.

518x4y6z10Rewrite radicand592x4y6x10Apply product rule for radicals592x4y6z10Simplify each square root532x2y3z5Rewrite and simplify coefficients15x2y3z52Simplified expression

Example 10.1.9

Simplify: 20x5y9z6. Assume all variables are positive.

Solution

We can apply the product rule for radicals to simplify by rewriting the variable’s exponent and rewrite the exponents so that one of the exponents is the largest even number.

20x5y9z6Rewrite radicand45x4xy8yz6Apply product rule for radicals45x4xy8yz6Simplify each square root25x2xy4yz3Rewrite and simplify coefficients2x2y4z35xySimplified expression

Simplify Radicals Homework

Simplify. Assume all variables are positive.

Exercise 10.1.1

245

Exercise 10.1.2

36

Exercise 10.1.3

12

Exercise 10.1.4

312

Exercise 10.1.5

6128

Exercise 10.1.6

8392

Exercise 10.1.7

192n

Exercise 10.1.8

196v2

Exercise 10.1.9

252x2

Exercise 10.1.10

100k4

Exercise 10.1.11

764x4

Exercise 10.1.12

536m

Exercise 10.1.13

45x2y2

Exercise 10.1.14

16x3y3

Exercise 10.1.15

320x4y4

Exercise 10.1.16

680xy2

Exercise 10.1.17

5245x2y3

Exercise 10.1.18

2180u3v

Exercise 10.1.19

8180x4y2z4

Exercise 10.1.20

280hj4k

Exercise 10.1.21

454mnp2

Exercise 10.1.22

125

Exercise 10.1.23

196

Exercise 10.1.24

338

Exercise 10.1.25

532

Exercise 10.1.26

7128

Exercise 10.1.27

763

Exercise 10.1.28

343b

Exercise 10.1.29

100n3

Exercise 10.1.30

200a3

Exercise 10.1.31

4175p4

Exercise 10.1.32

2128n

Exercise 10.1.33

8112p2

Exercise 10.1.34

72a3b4

Exercise 10.1.35

512a4b2

Exercise 10.1.36

512m4n3

Exercise 10.1.37

898mn

Exercise 10.1.38

272x2y2

Exercise 10.1.39

572x3y4

Exercise 10.1.40

650a4bc2

Exercise 10.1.41

32xy2z3

Exercise 10.1.42

832m2p4q

Write each expression in radical form with only positive exponents.

Exercise 10.1.43

m35

Exercise 10.1.44

(7x)32

Exercise 10.1.45

(10r)34

Exercise 10.1.46

(6b)43

Write each expression in exponential form.

Exercise 10.1.47

1(6x)3

Exercise 10.1.48

1(4n)7

Exercise 10.1.49

v

Exercise 10.1.50

5a

Evaluate without using a calculator.

Exercise 10.1.51

823

Exercise 10.1.52

432

Exercise 10.1.53

1614

Exercise 10.1.54

10032


This page titled 10.1: Simplify Radicals is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?