## SAT Test Prep

## CHAPTER 8

ESSENTIAL ALGEBRA I SKILLS

### Lesson 3: Working with Exponentials

**What Are Exponentials?**

An **exponential** is simply any term with these three parts:

If a term seems not to have a coefficient or exponent, the coefficient or exponent is always assumed to be 1!

**Examples:**

**Expand to Understand**

Good students memorize the rules for working with exponentials, but great students understand where the rules come from. They come from simply expanding the exponentials and then collecting or cancelling the factors.

**Example:**

What is (*x*^{5})^{2} in simplest terms? Do you *add* exponents to get *x*^{7}? *Multiply* to get *x*^{10}? *Power* to get *x*^{25}?

The answer is clear when you expand the exponential. Just remember that *raising to the* n*th power* simply means *multiplying by itself* n *times*. So . Doing this helps you to see and understand the rule of “multiplying the powers.”

**Adding and Subtracting Exponentials**

When adding or subtracting exponentials, you can combine only like terms, that is, terms with the same base and the same exponent. When adding or subtracting like exponentials, remember to leave the bases and exponents alone.

**Example:**

Notice that combining like terms always involves the Law of Distribution (__Chapter 7__, Lesson 2).

**Multiplying and Dividing Exponentials**

You can simplify a **product** or **quotient** of exponentials when the bases are the same or the exponents are the same.

If the bases are the same, add the exponents (when multiplying) or subtract the exponents (when dividing) and leave the bases alone.

If the exponents are the same, multiply (or divide) the bases and leave the exponents alone.

**Example:**

**Raising Exponentials to Powers**

When raising an exponential to a power, multiply the exponents, but don”t forget to raise the coefficient to the power and leave the base alone.

**Example:**

**Concept Review 3: Working with Exponentials**

__1.__ The three parts of an exponential are the __________, __________, and __________.

__2.__ When multiplying two exponentials *with the same base*, you should __________ the coefficients, __________ the bases, and __________ the exponents.

__3.__ When dividing two exponentials *with the same exponent*, you should __________ the coefficients, __________ the bases, and __________ the exponents.

__4.__ When multiplying two exponentials *with the same exponent*, you should __________ the coefficients, __________ the bases, and __________ the exponents.

__5.__ When dividing two exponentials *with the same base*, you should __________ the coefficients, __________ the bases, and __________ the exponents.

__6.__ To raise an exponential to a power, you should __________ the coefficient, __________ the base, and __________ the exponents.

Complete the tables:

Simplify, if possible.

**SAT Practice 3: Working with Exponentials**

**1**__.__ If , then

(A) –1

(E) 1

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

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

**3**__.__ If , then

(A) 1

(B) 3

(C) 6

(D) 9

(E) 12

**4**__.__ If , then

(A) 20

(B) 40

(C) 80

(D) 100

(E) 200

**5**__.__ If and *x* is positive, which of the following equals 5*y*^{2} in terms of *x*?

(A) 5^{2}*x*

(C) 25^{2}*x*

(D) 125^{2}*x*

**6**__.__ If , then

**7**__.__ If , then what is the effect on the value of *p* when *n* is multiplied by 4 and *m* is doubled?

(A) *p* is unchanged.

(B) *p* is halved.

(C) *p* is doubled.

(D) *p* is multiplied by 4.

(E) *p* is multiplied by 8.

**8**__.__ For all real numbers *n*,

(A) 2

(B) 2^{n}

(C) 2* ^{n}*1

**9**__.__ If *m* is a positive integer, then which of the following is equivalent to

(B) 3^{3}*m*

(D) 9^{m}

(E) 9^{3}*m*

**Answer Key 3: Working with Exponentials**

**Concept Review 3**

__1.__ coefficient, base, and exponent

__2.__ *multiply* the coefficients, *keep* the bases, and *add* the exponents.

__3.__ *divide* the coefficients, *divide* the bases, and *keep* the exponents.

__4.__ *multiply* the coefficients, *multiply* the bases, and *keep* the exponents.

__5.__ *divide* the coefficients, *keep* the bases, and *subtract* the exponents.

__6.__ *raise* the coefficient (to the power), *keep* the base, and *multiply* the exponents.

__7.__ –4* ^{x}* coefficient: –1; base: 4; exponent:

*x*

__8.__ (*xy*)^{–4} coefficient: –1; base: *xy;* exponent: –4

__9.__ *xy*^{–4} coefficient: *x;* base: *y;* exponent: –4

__10.__ (3*x*)^{9} coefficient: 1; base: 3*x;* exponent: 9

**SAT Practice 3**

__1.__ **B** You don”t need to plug in *g* = –4.1. Just simplify:

If

__7.__ **B** Begin by assuming

Then

If *n* is multiplied by 4 and *m* is doubled, then and , so which is half of the original value.

__8.__ **C** (Remember that 2* ^{n}* × 2

*equals 2*

^{n}^{2}

*n*, or 4

*, but*

^{n}