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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

Before you get started, take this readiness quiz.

  1. Simplify: y 5 y 4 .
    If you missed this problem, review [link] .
  2. Simplify: ( n 2 ) 6 .
    If you missed this problem, review [link] .
  3. Simplify: x 8 x 3 .
    If you missed this problem, review [link] .

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: n 2 n squared n 3 n cubed n 4 n to the fourth n 5 n 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 −5 to 5 . See [link] .

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.
First through fifth powers of integers from −5 to 5 .

Notice the signs in [link] . 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 n 2 = m , then n is a square root of m .

And we have used the notation m to denote the principal square root . So m 0 always.

We will now extend the definition to higher roots.

n Th root of a number

If b n = a , then b is an n th root of a number     a .

The principal n th root of a is written a n .

         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 b 3 , we use the term ‘cube root’ for a 3 .

We refer to [link] to help us find higher roots.

4 3 = 64 64 3 = 4 3 4 = 81 81 4 = 3 ( −2 ) 5 = −32 −32 5 = −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.

Properties of a n

When n is an even number and

  • a 0 , then a n is a real number
  • a < 0 , then a n is not a real number

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

Simplify: 8 3 81 4 32 5 .

Solution


8 3 Since ( 2 ) 3 = 8 . 2


81 4 Since ( 3 ) 4 = 81 . 3


32 5 Since ( 2 ) 5 = 32 . 2

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Simplify: 27 3 256 4 243 5 .

3 4 3

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Simplify: 1000 3 16 4 32 5 .

10 2 2

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Simplify: −64 3 −16 4 −243 5 .

Solution


  1. −64 3 Since ( −4 ) 3 = −64 . −4


  2. −16 4 Think, ( ? ) 4 = −16 . No real number raised to the fourth power is positive. Not a real number.


  3. −243 5 Since ( −3 ) 5 = −243 . −3
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Simplify: −125 3 −16 4 −32 5 .

−5 not real −2

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Simplify: −216 3 −81 4 −1024 5 .

−6 not real −4

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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 −64 3 = −4 .

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

Suppose we start with a = −5 .

( −5 ) 4 = 625 625 4 = 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.

Practice Key Terms 4

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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