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17.2: Simplify Expressions with Roots

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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

Before you get started, take this readiness quiz.

  1. Simplify: a. (9)2 b. 92 c. (9)3
    If you missed this problem, review Example 2.21.
  2. Round 3.846 to the nearest hundredth.
    If you missed this problem, review Example 1.34.
  3. Simplify: a. x3x3 b. y2y2y2 c. z3z3z3z3
    If you missed this problem, review Example 5.12.

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 'n2 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 a square root of 169

Definition 17.2.1: Square and Square Root of a Number

Square

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

Square Root

If n2=m, then n is a square root of 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 positive square root is also called the principal square root.

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 17.2.2: Square Root Notation

m is read "the square root of m."

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

radical signmradicand
Figure 8.1.1

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 17.2.1

Simplify:

  1. 144
  2. 289

Solution:

a.

144

Since 122=144.

12

b.

289

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

17

Exercise 17.2.1

Simplify:

  1. 64
  2. 225
Answer
  1. 8
  2. 15
Exercise 17.2.2

Simplify:

  1. 100
  2. 121
Answer
  1. 10
  2. 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 17.2.2

Simplify:

  1. 196
  2. 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

Exercise 17.2.3

Simplify:

  1. 169
  2. 81
Answer
  1. not a real number
  2. 9
Exercise 17.2.4

Simplify:

  1. 49
  2. 121
Answer
  1. 7
  2. not a real number

So far we have only talked about squares and square roots. Let’s now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

 We write:  We say: n2n squared n3n cubed n4n to the fourth power n5n to the fifth power 

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from 5 to 5. See Figure 8.1.2

The figure contains two tables. The first table has 9 rows and 5 columns. The first row is a header row with the headers “Number”, “Square”, “Cube”, “Fourth power”, and “Fifth power”. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the number 1 in each column. The fourth row contains the numbers 2, 4, 8, 16, 32. The fifth row contains the numbers 3, 9, 27, 81, 243. The sixth row contains the numbers 4, 16, 64, 256, 1024. The seventh row contains the numbers 5, 25, 125 625, 3125. The eighth row contains the expressions x, x squared, x cubed, x to the fourth power, and x to the fifth power. The last row contains the expressions x squared, x to the fourth power, x to the sixth power, x to the eighth power, and x to the tenth power. The second table has 7 rows and 5 columns. The first row is a header row with the headers “Number”, “Square”, “Cube”, “Fourth power”, and “Fifth power”. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the numbers negative 1, 1 negative 1, 1, negative 1. The fourth row contains the numbers negative 2, 4, negative 8, 16, negative 32. The fifth row contains the numbers negative 3, 9, negative 27, 81, negative 243. The sixth row contains the numbers negative 4, 16, negative 64, 256, negative 1024. The last row contains the numbers negative 5, 25, negative 125, 625, negative 3125.
Figure 8.1.2

Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of 2 to help you see this.

The image contains a table with 2 rows and 5 columns. The first row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row contains the numbers negative 2, 4, negative 8, 16, negative 32. Arrows point to the second and fourth columns with the label “Even power Positive result”. Arrows point to the first third and fifth columns with the label “Odd power Negative result”.
Figure 8.1.3

We will now extend the square root definition to higher roots.

Definition 17.2.3: Nth Root of a Number

If bn=a, then b is an nth root of a.

The principal nth root of a is written na.

The n is called the index of the radical.

Just like we use the word ‘cubed’ for b3, we use the term ‘cube root’ for 3a.

We can refer to Figure 8.1.2 to help find higher roots.

43=64364=434=81481=3(2)5=32532=2

Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Properties of na

When n is an even number and

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

When n is an odd number, na is a real number for all the values of a.

We will apply these properties in the next two examples.

Example 17.2.3

Simplify:

  1. 364
  2. 481
  3. 532

Solution:

a.

364

Since 43=64.

4

b.

481

Since (3)4=81.

3

c.

532

Since (2)5=32.

2

Exercise 17.2.5

Simplify:

  1. 327
  2. 4256
  3. 5243
Answer
  1. 3
  2. 4
  3. 3
Exercise 17.2.6

Simplify:

  1. 31000
  2. 416
  3. 5243
Answer
  1. 10
  2. 2
  3. 3

In this example be alert for the negative signs as well as even and odd powers.

Example 17.2.4

Simplify:

  1. 3125
  2. 416
  3. 5243

Solution:

a.

3125

Since (5)3=125.

5

b.

416

Think, (?)4=16. No real number raised to the fourth power is negative.

Not a real number.

c.

5243

Since (3)5=243.

3

Exercise 17.2.7

Simplify:

  1. 327
  2. 4256
  3. 532
Answer
  1. 3
  2. not real
  3. 2
Exercise 17.2.8

Simplify:

  1. 3216
  2. 481
  3. 51024
Answer
  1. 6
  2. not real
  3. 4

Estimate and Approximate Roots

When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the 4=2, what is the value of 21 or 350? In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.

To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of 11, we see 11 is between perfect square numbers 9 and 16, closer to 9. Its square root then will be between 3 and 4, but closer to 3.

The figure contains two tables. The first table has 5 rows and 2 columns. The first row is a header row with the headers “Number” and “Square Root”. The second row has the numbers 4 and 2. The third row is 9 and 3. The fourth row is 16 and 4. The last row is 25 and 5. A callout containing the number 11 is directed between the 9 and 16 in the first column. Another callout containing the number square root of 11 is directed between the 3 and 4 of the second column. Below the table are the inequalities 9 is less than 11 is less than 16 and 3 is less than square root of 11 is less than 4. The second table has 5 rows and 2 columns. The first row is a header row with the headers “Number” and “Cube Root”. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 91 is directed between the 64 and 125 in the first column. Another callout containing the number cube root of 91 is directed between the 4 and 5 of the second column. Below the table are the inequalities 64 is less than 91 is less than 125 and 4 is less than cube root of 91 is less than 5.
Figure 8.1.4

Similarly, to estimate 391, we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.

Example 17.2.5

Estimate each root between two consecutive whole numbers:

  1. 105
  2. 343

Solution:

a. Think of the perfect square numbers closest to 105. Make a small table of these perfect squares and their squares roots.

Table 8.1.1
  105
  .
Locate 105 between two consecutive perfect squares. 100<105<121
105 is between their square roots. 10<105<11

b. Similarly we locate 43 between two perfect cube numbers.

Table 8.1.2
  343
  .
Locate 43 between two consecutive perfect cubes. .
343 is between their cube roots. .
Exercise 17.2.9

Estimate each root between two consecutive whole numbers:

  1. 38
  2. 393
Answer
  1. 6<38<7
  2. 4<393<5
Exercise 17.2.10

Estimate each root between two consecutive whole numbers:

  1. 84
  2. 3152
Answer
  1. 9<84<10
  2. 5<3152<6

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the x key on your calculator. To find a cube root, or any root with higher index, you will use the yx key.

When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is and it is read ‘approximately’.

Suppose your calculator has a 10 digit display. You would see that

52.236067978 rounded to two decimal places is 52.24

4933.105422799 rounded to two decimal places is 4933.11

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

(2.236067978)2=5.000000002(3.105422799)4=92.999999991(2.24)2=5.0176(3.11)4=93.54951841

Their squares are close to 5, but are not exactly equal to 5. The fourth powers are close to 93, but not equal to 93.

Example 17.2.6

Round to two decimal places:

  1. 17
  2. 349
  3. 451

Solution:

a.

17

Use the calculator square root key.

4.123105626

Round to two decimal places.

4.12

174.12

b.

349

Use the calculator yx key.

3.659305710

Round to two decimal places.

3.66

3493.66

c.

451

Use the calculator yx key.

2.6723451177

Round to two decimal places.

2.67

4512.67

Exercise 17.2.11

Round to two decimal places:

  1. 11
  2. 371
  3. 4127
Answer
  1. 3.32
  2. 4.14
  3. 3.36
Exercise 17.2.12

Round to two decimal places:

  1. 13
  2. 384
  3. 498
Answer
  1. 3.61
  2. 4.38
  3. 3.15

Simplify Variable Expressions with Roots

The odd root of a number can be either positive or negative. For example,

Three equivalent expressions are written: the cube root of 4 cubed, the cube root of 64, and 4. There are arrows pointing to the 4 that is cubed in the first expression and the 4 in the last expression labeling them as “same”. Three more equivalent expressions are also written: the cube root of the quantity negative 4 in parentheses cubed, the cube root of negative 64, and negative 4. The negative 4 in the first expression and the negative 4 in the last expression are labeled as being the “same”.
Figure 8.1.13

But what about an even root? We want the principal root, so 4625=5.

But notice,

Three equivalent expressions are written: the fourth root of the quantity 5 to the fourth power in parentheses, the fourth root of 625, and 5. There are arrows pointing to the 5 in the first expression and the 5 in the last expression labeling them as “same”. Three more equivalent expressions are also written: the fourth root of the quantity negative 5 in parentheses to the fourth power in parentheses, the fourth root of 625, and 5. The negative 5 in the first expression and the 5 in the last expression are labeled as being the “different”.
Figure 8.1.14

How can we make sure the fourth root of 5 raised to the fourth power is 5? We can use the absolute value. |5|=5. So we say that when n is even nan=|a|. This guarantees the principal root is positive.

Definition 17.2.4: Simplifying Odd and Even Roots

For any integer n2,

when the index n is odd nan=a

when the index n is even nan=|a|

We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Example 17.2.7

Simplify:

  1. x2
  2. 3n3
  3. 4p4
  4. 5y5

Solution:

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

x2

Since the index n is even, nan=|a|.

b. This is an odd indexed root so there is no need for an absolute value sign.

3m3

Since the index is n is odd, nan=a.

m

c.

4p4

Since the index n is even nan=|a|.

|p|

d.

5y5

Since the index n is odd, nan=a.

y

Exercise 17.2.13

Simplify:

  1. b2
  2. 3w3
  3. 4m4
  4. 5q5
Answer
  1. |b|
  2. w
  3. |m|
  4. q
Exercise 17.2.14

Simplify:

  1. y2
  2. 3p3
  3. 4z4
  4. 5q5
Answer
  1. |y|
  2. p
  3. |z|
  4. q

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

Since (am)2=a2m.

(am)2

Since n is even nan=|a|.

|am|

So a2m=|am|.

We apply this concept in the next example.

Example 17.2.8

Simplify:

  1. x6
  2. y16

Solution:

a.

x6

Since (x3)2=x6.

(x3)2

Since the index n is even an=|a|.

|x3|

b.

y16

Since (y8)2=y16.

(y8)2

Since the index n is even nan=|a|.

y8

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

Exercise 17.2.15

Simplify:

  1. y18
  2. z12
Answer
  1. |y9|
  2. z6
Exercise 17.2.16

Simplify:

  1. m4
  2. b10
Answer
  1. m2
  2. |b5|

The next example uses the same idea for higher roots.

Example 17.2.9

Simplify:

  1. 3y18
  2. 4z8

Solution:

a.

3y18

Since (y6)3=y18.

3(y6)3

Since n is odd, nan=a.

y6

b.

4z8

Since (z2)4=z8.

4(z2)4

Since z2 is positive, we do not need an absolute value sign.

z2

Exercise 17.2.17

Simplify:

  1. 4u12
  2. 3v15
Answer
  1. |u3|
  2. v5
Exercise 17.2.18

Simplify:

  1. 5c20
  2. 6d24
Answer
  1. c4
  2. d4

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 positive.

Example 17.2.10

Simplify:

  1. 16n2
  2. 81c2

Solution:

a.

16n2

Since (4n)2=16n2.

(4n)2

Since the index n is even nan=|a|.

4|n|

b.

81c2

Since (9c)2=81c2.

(9c)2

Since the index n is even nan=|a|.

9|c|

Exercise 17.2.19

Simplify:

  1. 64x2
  2. 100p2
Answer
  1. 8|x|
  2. 10|p|
Exercise 17.2.20

Simplify:

  1. 169y2
  2. 121y2
Answer
  1. 13|y|
  2. 11|y|

This example just takes the idea farther as it has roots of higher index.

Example 17.2.11

Simplify:

  1. 364p6
  2. 416q12

Solution:

a.

364p6

Rewrite 64p6 as (4p2)3.

3(4p2)3

Take the cube root.

4p2

b.

416q12

Rewrite the radicand as a fourth power.

4(2q3)4

Take the fourth root.

2|q3|

Exercise 17.2.21

Simplify:

  1. 327x27
  2. 481q28
Answer
  1. 3x9
  2. 3|q7|
Exercise 17.2.22

Simplify:

  1. 3125q9
  2. 5243q25
Answer
  1. 5p3
  2. 3q5

The next examples have two variables.

Example 17.2.12

Simplify:

  1. 36x2y2
  2. 121a6b8
  3. 364p63q9

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

c.

364p63q9

Since (4p21q3)3=64p63q9

3(4p21q3)3

Take the cube root.

4p21q3

Exercise 17.2.23

Simplify:

  1. 100a2b2
  2. 144p12q20
  3. 38x30y12
Answer
  1. 10|ab|
  2. 12p6q10
  3. 2x10y4
Exercise 17.2.24

Simplify:

  1. 225m2n2
  2. 169x10y14
  3. 327w36z15
Answer
  1. 15|mn|
  2. 13|x5y7|
  3. 3w12z5

Access this online resource for additional instruction and practice with simplifying expressions with roots.

  • Simplifying Variables Exponents with Roots using Absolute Values

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.
  • nth Root of a Number
    • If bn=a, then b is an nth root of a.
    • The principal nth root of a is written na.
    • n is called the index of the radical.
  • Properties of na
    • When n is an even number and
      • a0, then na is a real number
      • a<0, then na is not a real number
    • When n is an odd number, na is a real number for all values of a.
  • Simplifying Odd and Even Roots
    • For any integer n2,
      • when n is odd nan=a
      • when n is even nan=|a|
    • We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Glossary

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

This page titled 17.2: Simplify Expressions with Roots is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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