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1.4.1: Radical Expressions

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Learning Objectives

By the end of this section, you will be able to:

  • Simplify expressions with roots
  • Estimate and approximate roots
  • Simplify variable expressions with roots
Be Prepared

Before you get started, take this readiness quiz.

1. Simplify

a. (9)2

b. 92

2. Round 3.846 to the nearest hundredth.

3. Simplify

a. x3x3

b. y2y2

 

In this section we deal with radical expressions of index 2 called square roots.

Simplify Expressions with Roots

In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write n2 and read it 'n squared’. This number is called the square of n, and n is called the square root. For example,

132 is read "13 squared"

169 is called the square of 13, since 132=169

13 is called a square root of 169

Definition 1.4.1.1

Square

If n2=m, then m is the square of n.

Square Root

If n2=m, then n is a square root of m.

In words,

a square root of m is a number whose square is m.

Notice (13)2=169 also, so 13 is also a square root of 169. Therefore, both 13 and 13 are square roots of 169.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write, m, which denotes the positive square root of m. The non-negative square root is also called the principal square root. This is the square root approximated by using the root symbol of your calculator!

We also use the radical sign for the square root of zero. Because 02=0,0=0. Notice that zero has only one square root.

Definition 1.4.1.2

m is read "the square root of m."

If n2=m, then n=m, for n0.

radical signmradicand
 

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 169=13. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 169=13.

Example 1.4.1.3

Simplify:

a. 144

b. 289

Solution

a.

144

Since 122=144, and 120

12

b.

289

Since 172=289, 170, and the negative is in front of the radical sign.

17

Try It 1.4.1.4

Simplify:

a. 64

b. 225

Answer

a. 8

b. 15

Try It 1.4.1.5

Simplify:

a. 100

b. 121

Answer

a. 10

b. 11

Can we simplify 49? Is there a number whose square is 49?

(___)2=49

Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to 49. The square root of a negative number is not a real number.

Example 1.4.1.6

Simplify:

a. 196

b. 64

Solution

a.

196

There is no real number whose square is 196.

196 is not a real number.

b.

64

The negative is in front of the radical.

8

Try It 1.4.1.7

Simplify:

a. 169

b. 81

Answer

a. not a real number

b. 9

Try It 1.4.1.8

Simplify:

a. 49

b. 121

Answer

a. 7

b. not a real number

Properties of a

When

  • a0, then a is a real number.
  • a<0, then a is not a real number.

Simplify Variable Expressions with Square Roots

Note, for example,

42=16=4

but,

(4)2=16=4,

So that the result is positive.

How can we make sure the square root of 5 squared is 5? We can use the absolute value. |5|=5: a2=|a|. This guarantees the principal root is positive.

Note that the 'root button' and the 'square button' are the same on most calculators and if a is positive, applying the square button and then the root button (or vice versa) will result in the return of a.

Summary

We have

a2=|a|

Example 1.4.1.9

Simplify x2.

Solution

We use the absolute value to be sure to get the positive root.

x2=|x|

Try It 1.4.1.10

Simplify b2.

Answer

|b|

What about square roots of higher powers of variables? The power property of exponents says (am)n=amn. So if we square am, the exponent will become 2m.

(am)2=a2m

Looking now at the square root.

a2m=(am)2

Since 2 is even, 2x2=|x|. So

a2m=|am|.

We apply this concept in the next example.

Example 1.4.1.11

Simplify:

a. x6

b. y16

Solution

a.

x6

Since (x3)2=x6, this is equal to

(x3)2.

Since a2=|a|, this is equal to

|x3|

b.

y16

Since (y8)2=y16, this is equal to

(y8)2.

Since a2=|a|, this is equal to

y8

In this case the absolute value sign is not needed as y8 is positive.

Try It 1.4.1.12

Simplify:

a. y18

b. z12

Answer

a. |y9|

b. z6

Try It 1.4.1.13

Simplify:

a. m4

b. b10

Answer

a. m2

b. |b5|

Note that if the variables are positive, then the exponent gets halved which is the same action you would do to simplify ()12 if rational exponents followed the same rules as integer exponents. For example, if x is non-negative,

x6=(x6)12=x612=x3.

We will treat this in greater detail a little later in this chapter.

In the next example, we now have a coefficient in front of the variable. The concept a2m=|am| works in much the same way.

16r22=4|r11| because (4r11)2=16r22.

But notice 25u8=5u4 and no absolute value sign is needed as u4 is always non-negative.

Example 1.4.1.14

Simplify:

a. 16n2

b. 81c2

Solution

a.

16n2

Since (4n)2=16n2, this is equal to

(4n)2.

Since a2=|a|, this is equal to

4|n|.

b.

81c2

Since (9c)2=81c2, this is equal to

(9c)2.

Since a2=|a|, this is then equal to

9|c|.

Try It 1.4.1.15

Simplify:

a. 64x2

b. 100p2

Answer

a. 8|x|

b. 10|p|

Try It 1.4.1.16

Simplify:

a. 169y2

b. 121y2

Answer

a. 13|y|

b. 11|y|

The next examples have two variables.

Example 1.4.1.17

Simplify:

a. 36x2y2

b. 121a6b8

Solution

a.

36x2y2

Since (6xy)2=36x2y2

(6xy)2

Take the square root.

6|xy|

b.

121a6b8

Since (11a3b4)2=121a6b8

(11a3b4)2

Take the square root.

11|a3|b4

Try It 1.4.1.18

Simplify:

a. 100a2b2

b. 144p12q20

Answer

a. 10|ab|

b. 12p6q10

Try It 1.4.1.19

Simplify:

a. 225m2n2

b. 169x10y14

Answer

a. 15|mn|

b. 13|x5y7|

Writing Exercises 1.4.1.20
  1. What is a square root?
  2. Explain why 9=3.
  3. What happens if we square a square root?
  4. What is the index? Radicand?
Exit Problem 1.4.1.21

Simplify 81a4b6.

Key Concepts

  • Square Root Notation
    • m is read ‘the square root of m
    • If n2=m, then n=m, for n0.
      The image shows the variable m inside a square root symbol. The symbol is a line that goes up along the left side and then flat above the variable. The symbol is labeled “radical sign”. The variable m is labeled “radicand”.
      Figure 8.1.1
    • The square root of m, m, is a positive number whose square is m.
  • Properties of a
    • a0, then a is a real number
    • a<0, then a is not a real number
  • Simplifying Odd and Even Roots
    • a2=|a|. We must use the absolute value signs when we take a square root of an expression with a variable in the radical.

This page titled 1.4.1: Radical Expressions is shared under a CC BY license and was authored, remixed, and/or curated by Holly Carley and Ariane Masuda.

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