SAT Mc Graw Hill 2011

CHAPTER 8 / ESSENTIAL ALGEBRA I SKILLS 309

Lesson 3: Working with Exponentials

What Are Exponentials?

An exponentialis simply any term with these three
parts:

If a term seems not to have a coefficient or ex-

ponent, the coefficient or exponent is always

assumed to be 1!

Examples:
2 xmeans 2x^1 y^3 means 1y^3

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 addex-
ponents to get x^7? Multiplyto get x^10? Powerto
get x^25?

The answer is clear when you expand the exponential.
Just remember that raising to thenth powersimply
means multiplying by itselfn times.So (x^5 )^2 =(x^5 )(x^5 )
=(x
x
x
x
x)(x
x
x
x
x) =x^10. 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, re-

member to leave the bases and exponents alone.

Example:
5 x^3 + 6 x^3 + 4 x^2 =(5x^3 + 6 x^3 ) + 4 x^2 =x^3 (5 +6) + 4 x^2 =
11 x^3 + 4 x^2

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

Multiplying and Dividing Exponentials

You can simplify a productor quotientof exponen-

tials 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.

(5m^5 )(12m^2 ) =(5)(m)(m)(m)(m)(m)(12) (m)(m)

=(5)(12)(m)(m)(m)(m)(m)(m)(m)

= 60 m^7

If the exponents are the same, multiply (or divide)

the bases and leave the exponents alone.

Example:

(3m^4 )(7n^4 ) =(3)(m)(m)(m)(m)(7)(n)(n)(n)(n)

=(3)(7)(mn)(mn)(mn)(mn)

=21(mn)^4

Raising Exponentials to Powers

When raising an exponential to a power, multi-

ply the exponents, but don’t forget to raise the

coefficient to the power and leave the base alone.

Example:

(3y^4 )^3 =(3y^4 )(3y^4 )(3y^4 )

=(3y
y
y
y)(3y
y
y
y)(3y
y
y
y)

=(3)(3)(3)(y
y
y
y)(y
y
y
y)(y
y
y
y)

= 27 y^12

512

3

51212121212

3333

5

5

()

()

=

()()()()()

()()()())( )

=

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

⎛

⎝⎜

⎞

⎠⎟

3

5

12

3

12

3

12

3

12

3

112

3

545

⎛

⎝⎜

⎞

⎠⎟

= ()

6

3

6

3

7

4

p

p

ppppppp

pppp

=

()()()()()()()

()()()()

= 22 p^3

Coefficient

Base

Exponent

4 x

3