Basic Mathematics for College Students

In Example 6, part a, was raised to an even power, and the answer was positive.

In parts b and c, and were raised to odd powers, and, in each case, the answer

was negative. These results suggest a general rule.

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 2

170 Chapter 2 The Integers

Solution

a.We read as “negative two raised to the fourth power” or as “the fourth

power of negative two.” Note that the exponent is even.

Write the base, 2, as a factor 4 times.

Multiply the first two negative factors and the last two negative

factors to produce positive products.

The result is positive.

b.We read as “negative five raised to the third power” or as “the third

power of negative five,” or as “ negative five, cubed.” Note that the exponent

is odd.

Write the base, 5, as a factor 3 times.

Multiply the first two negative factors to produce a

positive product.

The result is negative.

c.We read as “negative one raised to the fifth power” or as “the fifth

power of negative one.” Note that the exponent is odd.

Write the base, 1, as a factor 5 times.

Multiply the first and second negative factors and multiply the

third and fourth negative factors to produce positive products.

 1 The result is negative.

1(1)(1)

(1)^5 (1)(1)(1)(1)(1)

(1)^5

 125

 25 (5)

(5)^3 (5)(5)(5)

(5)^3

 16

4(4)

(2)^4 (2)(2)(2)(2)

(2)^4

Even and Odd Powers of a Negative Integer

When a negative integer is raised to an even power, the result is positive.

When a negative integer is raised to an odd power, the result is negative.

Although the exponential expressions and look similar, they are not

the same. We read as “negative 3 squared” and as “the opposite of the

square of three.” When we evaluate them, it becomes clear that they are not

equivalent.

Different results

 9  9

Since there are no

parentheses around

3, the base is 3.

The exponent is 2.

Because of the ^32 (33)

parentheses, the

base is 3. The

exponent is 2.

(3)^2 (3)(3)

(3)^2  32

(3)^2  32

Now TryProblems 61, 65, and 67

Caution! The base of an exponential expression does not includethe

negative sign unless parentheses are used.

Positive base: 7 Negative base:  7

 73 (7)^3

 

  

V

2

2

5

 5

125