## SAT Test Prep

## CHAPTER 11

ESSENTIAL ALGEBRA 2 SKILLS

### Lesson 6: Negative and Fractional Exponents

**Exponents Review**

In __Chapter 8__, Lesson 3, we discussed the practical definition of exponentials:

The expression *x ^{n}* means

*x*multiplied by itself

*n*times.

This is a useful definition when you need to evaluate something like 4^{3}: you simply multiply 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 expressions, you must expand your understanding of exponents.

**Zero and Negative Exponents**

Using what you have learned in Lesson 1 of this chapter, what are the next three terms of this sequence?

The rule seems to be “divide by 3,” so the next three terms are 1, , and .

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

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, , , , and . This means that the pattern can help us to understand zero and negative exponents: , , and . 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:

Notice that raising a positive number to a negative power *does not* produce a negative result. For instance 3^{–2} does not equal –9; it equals .

**Fractional Exponents**

What if a number is raised to a *fractional* exponent? For instance, what does 8^{1/3} mean? 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 .

Using the rule above, . In other words, when you raise 8^{1/3} to the 3rd power, the result is 8. This means that 8^{1/3} is the same as the cube root of 8, and, in general:

The expression *x*^{1/n} means , or the *n*th root of *x*. For example, can be written as *a*^{1/2}.

**Example:**

What is the value of 16^{3/4}?

The first step is to see that 16^{3/4} is the same as (16^{1/4})^{3}(because . Using the definition above, 16^{1/4} is the 4th root of 16, which is 2 (because ). So .

The expression *x ^{m}*/

*n*means the

*n*th root of

*x*raised to the

*m*th power. For instance, 4

^{3/2}means the square root of 4 raised to the third power, or .

**Concept Review 6: Negative and Fractional Exponents**

Evaluate the following expressions without a calculator.

__1.__ 5^{–2}

__2.__ 9^{1/2}

__3.__ 2^{–5}

__4.__ 25^{–1/2}

__5.__ 4^{3/2}

Simplify the following expressions, eliminating any negative or fractional exponents.

__7.__ *x*^{1/3}

__8.__ (4*g*)^{1/2}

__9.__ 4*x*^{–2}

__10.__ (4*y*)^{–2}

__11.__ (9*m*)^{3/2}

__12.__ (27*b*)^{1/3}/(9*b*)^{–1/2}

__13.__ If , what is the value of *x*?

__14.__ If , what is the value of *b*?

__15.__ If , what is the value of 2^{3}*m*?

**SAT Practice 6: Negative and Fractional Exponents**

**1**__.__ If , then what is the value of 4^{–}* ^{n}*?

**2**__.__ If , then

(A) –5^{2}

(B) 5^{–2}

(D) 5^{1/2}

**3**__.__ If , then *m* =

**4**__.__ For all values of *n*,

(A) 3

(C) 3^{n}

(D) 9^{2}

(E) 9^{n}

**5**__.__ If *x* is a positive number, then

(A) *x*^{3/4}

(B) *x*^{–(1/4)}

(C) *x*^{3/4}

**6**__.__ If and *x* is positive, then

(A) *x*^{1/5}

(B) *x*^{1/8}

(C) *x*^{8/15}

(D) *x*^{8}

(E) *x*^{15}

**Answer Key 6: Negative and Fractional Exponents**

**Concept Review 6**

**SAT Practice 6**

Perhaps a simpler method is to simply pick *n* to be 0 (because *n* can be any number). This gives . The only choice that equals 3 when is (A).

(Remember the quick way to add fractions: “zip-zap-zup” from __Chapter 7__, Lesson 3.)