Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

9.7: Higher Roots

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

Learning Objectives

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

  • Simplify expressions with higher roots
  • Use the Product Property to simplify expressions with higher roots
  • Use the Quotient Property to simplify expressions with higher roots
  • Add and subtract higher roots
Note
  1. Simplify: y5y4.
    If you missed this problem, review Example 6.2.7.
  2. Simplify: (n2)6.
    If you missed this problem, review Example 6.2.19.
  3. Simplify: x8x3.
    If you missed this problem, review Example 6.5.1.

Simplify Expressions with Higher Roots

Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

We write:We say:n2n squaredn3n cubedn4n to the fourthn5n to the fifth

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 −5to5. See Figure \PageIdnex1.

This figure consists of two tables. The first table shows the results of raising the numbers 1, 2, 3, 4, 5, x, and x squared to the second, third, fourth, and fifth powers. The second table shows the results of raising the numbers negative one through negative five to the second, third, fourth, and fifth powers. The table first has five columns and nine rows. The second has five columns and seven rows. The columns in both tables are labeled, “Number,” “Square,” “Cube,” “Fourth power,” “Fifth power,” nothing,  “Number,” “Square,” “Cube,” “Fourth power,” and “Fifth power.” In both tables, the next row reads: n, n squared, n cubed, n to the fourth power, n to the fifth power, nothing, n, n squared, n cubed, n to the fourth power, and n to the fifth power. In the first table, 1 squared, 1 cubed, 1 to the fourth power, and 1 to the fifth power are all shown to be 1. In the next row, 2 squared is 4, 2 cubed is 8, 2 to the fourth power is 16, and 2 to the fifth power is 32. In the next row, 3 squared is 9, 3 cubed is 27, 3 to the fourth power is 81, and 3 to the fifth power is 243. In the next row, 4 squared is 16, 4 cubed is 64, 4 to the fourth power is 246, and 4 to the fifth power is 1024. In the next row, 5 squared is 25, 5 cubed is 125, 5 to the fourth power is 625, and 5 to the fifth power is 3125. In the next row, x squared, x cubed, x to the fourth power, and x to the fifth power are listed. In the next row, x squared squared is x to the fourth power, x cubed squared is x to the fifth power, x squared to the fourth power is x to the eighth power, and x squared to the fifth power is x to the tenth power. In the second table, negative 1 squared is 1, negative 1 cubed is negative 1, negative 1 to the fourth power is 1, and negative 1 to the fifth power is negative 1. In the next row, negative 2 squared is 4, negative 2 cubed is negative 8, negative 2 to the fourth power is 16, and negative 2 to the fifth power is negative 32. In the next row, negative 4 squared is 16, negative 4 cubed is negative 64, negative 4 to the fourth power is 256, and negative 4 to the fifth power is negative 1024. In the next row, negative 5 squared is 25, negative 5 cubed is negative 125, negative 5 to the fourth power is 625, and negative 5 to the fifth power is negative 3125.
Figure 9.7.1: First through fifth powers of integers from −5 to 5.

Notice the signs in Figure 9.7.1. 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 below to help you see this.

This figure has five columns and two rows. The first row labels each column: n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row reads: negative 2, 4, negative 8, 16, and negative 32.

Earlier in this chapter we defined the square root of a number.

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

And we have used the notation m to denote the principal square root. So m0 always.

We will now extend the definition to higher roots.

Definition: NTH ROOT OF A NUMBER

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

The principal nth root of a is written na=b

n is called the index of the radical.

We do not write the index for a square root. Just like we use the word ‘cubed’ for b3, we use the term ‘cube root’ for 3a.

We refer to Figure 9.7.1 to help us find higher roots.

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

Could we have an even root of a negative number? No. 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.

Definition: 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.

Example 9.7.1

Simplify:

  1. 38
  2. 481
  3. 532.
Answer
1. 38
Since (2)3=8. 2
2. 481
Since (3)4=81. 3
3. 532
Since (2)5=32. 2
Example 9.7.2

Simplify:

  1. 327
  2. 4256
  3. 5243.
Answer
  1. 3
  2. 4
  3. 3
Example 9.7.3

Simplify:

  1. 31000
  2. 416
  3. 532.
Answer
  1. 10
  2. 2
  3. 2
Example 9.7.4

Simplify:

  1. 364
  2. 416
  3. 5243.
Answer
1. 364
Since (4)3=64. −4
2. 416
Think, (?)4=16.No real number raised to the fourth power is positive. Not a real number.
3. 5243
Since (3)5=243. −3
Example 9.7.5

Simplify:

  1. 3125
  2. 416
  3. 532.
Answer
  1. −5
  2. not real
  3. −2
Example 9.7.6

Simplify:

  1. 3216
  2. 481
  3. 51024.
Answer
  1. −6
  2. not real
  3. −4
When we worked with square roots that had variables in the radicand, we restricted the variables to non-negative values. Now we will remove this restriction.

The odd root of a number can be either positive or negative. We have seen that 364=4.

But the even root of a non-negative number is always non-negative, because we take the principal nth root.

Suppose we start with a=−5.

(5)4=6254625=5

How can we make sure the fourth root of −5 raised to the fourth power, (5)4 is 5? We will see in the following property.

Definition: SIMPLIFYING ODD AND EVEN ROOTS

For any integer n2,

when n is oddnan=awhen n is evennan=|a|

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

Example 9.7.7

Simplify:

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

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

1. x2
Since (x)2=x2 and we want the positive root. |x|
2. 3n3
Since (n)3=n3. It is an odd root so there is no need for an absolute value sign. n
3. 4p4
Since (p)4=p4 and we want the positive root. |p|
4. 5y5
Since (y)5=y5. It is an odd root so there is no need for an absolute value sign. y
Example 9.7.8

Simplify:

  1. b2
  2. 3w3
  3. 4m4
  4. 5q5.
Answer
  1. |b|
  2. w
  3. |m|
  4. q
Example 9.7.9

Simplify:

  1. y2
  2. 3p3
  3. 4z4
  4. 5q5
Answer
  1. |y|
  2. p
  3. |z|
  4. q
Example 9.7.10

Simplify:

  1. 3y18
  2. 4z8.
Answer
1. 3y18
Since (y6)3=y18. 3(y6)3
  y6
2. 4z8
Since (z2)4=z8. 4(z2)4
Since z2 is positive, we do not need an absolute value sign. z2
Example 9.7.11

Simplify:

  1. 4u12
  2. 3v15.
Answer
  1. u3
  2. v5
Example 9.7.12

Simplify:

  1. 5c20
  2. 6d24.
Answer
  1. c4
  2. d4
Example 9.7.13

Simplify:

  1. 364p6
  2. 416q12.
Answer
1. 364p6
Rewrite 64p6 as (4p2)3. 3(4p2)3
Take the cube root. 4p2
2. 416q12
Rewrite the radicand as a fourth power. 4(2q3)4
Take the fourth root. 2|q3|
Example 9.7.14

Simplify:

  1. 327x27
  2. 481q28.
Answer
  1. 3x9
  2. 3q7
Example 9.7.15

Simplify:

  1. 3125p9
  2. 5243q25
Answer
  1. 5p3
  2. 3q5

Use the Product Property to Simplify Expressions with Higher Roots

We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An nth root is considered simplified if it has no factors of mn.

Definition: SIMPLIFIED NTH ROOT

na is considered simplified if a has no factors of mn.

We will generalize the Product Property of Square Roots to include any integer root n2.

Definition: PRODUCT PROPERTY OF NTH ROOTs

nab=na·nb and na·nb=nab

when na and nb are real numbers and for any integer n2

Example 9.7.16

Simplify:

  1. 3x4
  2. 4x7.
Answer

1.

3x4
Rewrite the radicand as a product using the largest perfect cube factor. 3x3·x
Rewrite the radical as the product of two radicals. 3x3·3x
Simplify. x3x
2. 4x7
Rewrite the radicand as a product using the greatest perfect fourth power factor. 4x4·x3
Rewrite the radical as the product of two radicals. 4x4·4x3
Simplify. |x|4x3
Example 9.7.17

Simplify:

  1. 4y6
  2. 3z5.
Answer
  1. |y4y2
  2. z3z2
Example 9.7.18

Simplify:

  1. 5p8
  2. 6q13.
Answer
  1. p5p3
  2. q26q
Example 9.7.19

Simplify:

  1. 316
  2. 4243.
Answer
1. 316
  324
Rewrite the radicand as a product using the largest perfect cube factor. 323·2
Rewrite the radical as the product of two radicals. 323·32
Simplify. 232
2. 4243
  435
Rewrite the radicand as a product using the greatest perfect fourth power factor. 434·3
Rewrite the radical as the product of two radicals. 434·43
Simplify. 343
Example 9.7.20

Simplify:

  1. 381
  2. 464.
Answer
  1. 333
  2. 244
Example 9.7.21

Simplify:

  1. 3625
  2. 4729.
Answer
  1. 535
  2. 349

Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.

Example 9.7.22

Simplify:

  1. 324x7
  2. 480y14.
Answer
1. 324x7
Rewrite the radicand as a product using perfect cube factors. 323x6·3x
Rewrite the radical as the product of two radicals. 323x6·33x
Rewrite the first radicand as (2x2)3 3(2x2)3·33x
Simplify. 2x233x
2. 480y14
Rewrite the radicand as a product using perfect fourth power factors. 424y12·5y2
Rewrite the radical as the product of two radicals. 424y12·45y2
Rewrite the first radicand as (2y3)4 4(2y3)4·45y2
Simplify. 2|y3|45y2
Example 9.7.23

Simplify:

  1. 354p[10]
  2. 464q10.
Answer
  1. 3p332p
  2. 2q244q2
Example 9.7.24

Simplify:

  1. 3128m11
  2. 4162n7.
Answer
  1. 4m332m2
  2. 3|n|42n3
Example 9.7.25

Simplify:

  1. 327
  2. 416.
Answer
1. 327
Rewrite the radicand as a product using perfect cube factors. 3(3)3
Take the cube root. −3
2. 416
There is no real number n where n4=16. Not a real number.
Example 9.7.26

Simplify:

  1. 3108
  2. 448.
Answer
  1. 334
  2. not real
Example 9.7.27

Simplify:

  1. 3625
  2. 4324.
Answer
  1. 535
  2. not real

Use the Quotient Property to Simplify Expressions with Higher Roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.

Example 9.7.28

Simplify:

  1. 3a8a5
  2. 4a10a2.
Answer

1.

3a8a5
Simplify the fraction under the radical first. 3a3
Simplify. a
2. 4a10a2
Simplify the fraction under the radical first. 4a8
Rewrite the radicand using perfect fourth power factors. 4(a2)4
Simplify. a2
Example 9.7.29

Simplify:

  1. 4x7x3
  2. 4y17y5.
Answer
  1. |x|
  2. y3
Example 9.7.30

Simplify:

  1. 3m13m7
  2. 5n12n2.
Answer
  1. m2
  2. n2

Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.

Definition: QUOTIENT PROPERTY OF NTH ROOTS

nab=nanb and nanb=nab

when na and nb are real numbers, b0, and for any integer n2

Exercise 9.7.31

Simplify:

  1. 310832
  2. 496x743x2
Answer
1. 310832
Neither radicand is a perfect cube, so use the Quotient Property to write as one radical. 31082
Simplify the fraction under the radical. 354
Rewrite the radicand as a product using perfect cube factors. 3(3)3·2
Rewrite the radical as the product of two radicals. 3(3)3·32
Simplify. 332
2. 496x743x2
Neither radicand is a perfect fourth power, so use the Quotient Property to write as one radical 496x73x2
Simplify the fraction under the radical. 432x5
Rewrite the radicand as a product using perfect fourth power factors. 424x4·2x
Rewrite the radical as the product of two radicals. 4(2x)4·42x
Simplify. 2|x|42x
Example 9.7.32

Simplify:

  1. 353232
  2. 4486m1143m5
Answer
  1. not real
  2. 3|m|42m2
Example 9.7.33

Simplify:

  1. 319233
  2. 4324n742n3.
Answer
  1. −4
  2. 3|n|42

If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.

Example 9.7.34

Simplify:

  1. 324x7y3
  2. 448x10y8.
Answer
1. 324x7y3
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals. 324x73y3
Rewrite each radicand as a product using perfect cube factors. 38x6·3x3y3
Rewrite the numerator as the product of two radicals. 3(2x2)3·33x3y3
Simplify. 2x233xy
2. 448x10y8
The fraction in the radicand cannot be simplified. Use the Quotient Property to write as two radicals. 448x104y8
Rewrite each radicand as a product using perfect cube factors. 416x8·3x24y8
Rewrite the numerator as the product of two radicals. 4(2x2)4·43x24(y2)4
Simplify. 2x243x2y2
Example 9.7.35

Simplify:

  1. 3108c10d6
  2. 480x10y5.
Answer
  1. 3c334cd2
  2. x2y480x2y
Example 9.7.36

Simplify:

  1. 340r3s
  2. 4162m14n12
Answer
  1. r340s
  2. 3m342m2n3

Add and Subtract Higher Roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals.

Definition: LIKE RADICALS

Radicals with the same index and same radicand are called like radicals.

Like radicals have the same index and the same radicand.

  • 9442x and 2442x are like radicals.
  • 53125x and 63125y are not like radicals. The radicands are different.
  • 251000q and 441000q are not like radicals. The indices are different.

We add and subtract like radicals in the same way we add and subtract like terms. We can add 9442x+(2442x) and the result is 7442x.

Example 9.7.37

Simplify:

  1. 34x+34x
  2. 448248
Answer
1. 34x+34x
The radicals are like, so we add the coefficients 234x
2. 448248
The radicals are like, so we subtract the coefficients. 248
Example 9.7.38

Simplify:

  1. 53x+53x
  2. 33939
Answer
  1. 253x
  2. 239
Example 9.7.39

Simplify:

  1. 410y+410y
  2. 56323632.
Answer
  1. 2410y
  2. 2632

When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.

Example 9.7.40

Simplify:

  1. 354316
  2. 448+4243.
Answer
1. 354316
Rewrite each radicand using perfect cube factors. 327·3238·32
Rewrite the perfect cubes. 3(3)3·323(2)3·32
Simplify the radicals where possible. 332232
Combine like radicals. 32
2. 448+4243
Rewrite using perfect fourth power factors. 416·43+481·43
Rewrite each radicand as a product using perfect cube factors. 4(2)4·43+4(3)4·43
Rewrite the numerator as the product of two radicals. 243+343
Simplify. 543
Example 9.7.41

Simplify:

  1. 3192381
  2. 432+4512.
Answer
  1. 33
  2. 642
Example 9.7.42

Simplify:

  1. 31083250
  2. 564+5486.
Answer
  1. 32
  2. 552
Example 9.7.43

Simplify:

  1. 324x4381x7
  2. 4162y9+4512y5.
Answer
1. 324x4381x7
Rewrite each radicand using perfect cube factors. 38x3·33x327x6·33x
Rewrite the perfect cubes. 3(2x)3·33x3(3x2)3·33x
Simplify the radicals where possible. 2x33x(3x233x)
2. 4162y9+4516y5
Rewrite using perfect fourth power factors. 481y8·42y+4256y4·42y
Rewrite each radicand as a product using perfect cube factors. 4(3y2)4·42y+4(4y)4·42y
Rewrite the numerator as the product of two radicals. 3y242y+4|y|42y
Example 9.7.44

Simplify:

  1. 332y53108y8
  2. 4243r11+4768r10.
Answer
  1. 2y34y2+3y234y2
  2. 3r243r3+4r243r2
Example 9.7.45

Simplify:

  1. 340z73135z4
  2. 480s13+41280s6.
Answer
  1. 2z235z+3z535z
  2. 2s345s+4|s|45s
Access these online resources for additional instruction and practice with simplifying higher roots.
  • Simplifying Higher Roots
  • Add/Subtract Roots with Higher Indices

Key Concepts

  • 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.
    • For any integer n2, when n is odd nan=a
    • For any integer n2, when n is even nan=|a|
  • na is considered simplified if a has no factors of mn.
  • nab=na·nb and na·nb=nab
  • nab=nanb and nanb=nab
  • To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.

Glossary

nth root of a number
If bn=a, then b is an nth root of a.
principal nth root
The principal nth root of a is written na.
index
na n is called the index of the radical.
like radicals
Radicals with the same index and same radicand are called like radicals.

This page titled 9.7: Higher Roots is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

Support Center

How can we help?