SAT Mc Graw Hill 2011

CHAPTER 11 / ESSENTIAL ALGEBRA 2 SKILLS 433

Lesson 6: Negative and Fractional Exponents

Exponents Review

In Chapter 8, Lesson 3, we discussed the practical de-

finition of exponentials:

The expression xnmeans xmultiplied by itself

ntimes.

This is a useful definition when you need to evaluate

something like 4^3 : you simply multiply 4 × 4 ×4 and

get 64. But what about expressions like 4^0 or 4−^3 or

4 1/2? How do you multiply 4 by itself 0 times, or − 3

times, or half of a time? It doesn’t make much sense

to think of it that way. So to understand such ex-

pressions, you must expand your understanding of

exponents.

Zero and Negative Exponents

Using what you have learned in Lesson 1 of this chap-

ter, what are the next three terms of this sequence?

81, 27, 9, 3, _____, _____, _____

The rule seems to be “divide by 3,” so the next three

terms are 1,^1 ⁄ 3 , and^1 ⁄ 9.

Now, what are the next three terms of this sequence?

34 , 3^3 , 3^2 , 3^1 , _____, _____, _____

Here, the rule seems to be “reduce the power by 1,” so

that the next three terms are 3^0 , 3−^1 , and 3−^2.

Notice that the two sequences are exactly the same,

that is, 3^4 =81, 3^3 =27, 3^2 =9, and 3^1 =3. This means

that the pattern can help us to understand zero and

negative exponents: 3^0 =1, 3−^1 =^1 ⁄ 3 , and 3−^2 =^1 ⁄ 9. Now,

here’s the million-dollar question:

Without a calculator, how do you write 3−^7 without a

negative exponent?

If you follow the pattern you should see that

and, in general:

x^0 1 and

Notice that raising a positive number to a neg-

ative power does notproduce a negative result.

For instance 3−^2 does not equal −9; it equals^1 ⁄ 9.

Fractional Exponents

What if a number is raised to a fractionalexponent?

For instance, what does 81/3mean? To understand

expressions like this, you have to use the basic rules

of exponents from Chapter 8, Lesson 3. Specifically,

you need to remember that xnxmxm+n.

(81/3)^3 = 8 1/3× 8 1/3× 8 1/3. Using the rule above, 81/3× 8 1/3×

8 1/3= 8 1/3 +1/3 +1/3= 81 =8. In other words, when you raise

8 1/3to the 3rd power, the result is 8. This means that

8 1/3is the same as the cube root of 8, and, in general:

The expressionx1/nmeans , or the nth root

of x. For example, can be written as a1/2.

Example:

What is the value of 163/4?

The first step is to see that 163/4is the same as (161/4)^3

(because (161/4)^3 = 16 1/4× 16 1/4× 16 1/4= 16 3/4). Using the

definition above, 161/4is the 4th root of 16, which is 2

(because 2^4 =16). So (161/4)^3 = 23 =8.

The expression xm/nmeans the nth root of x

raised to the mth power. For instance, 43/2

means the square root of 4 raised to the third

power, or 2^3 =8.

a

nx

x

x

n

n

− =^1

3

1

3

7

7

− =