Algebra Demystified 2nd Ed

96 algebra De mystif ieD

DeMYSTiFieD / algebra DeMYSTiFieD / HuttenMuller / 000-0 / Chapter 5

The expression 4x is shorthand for the sum x + x + x + x, that is, x added to

itself four times. Likewise x^4 is shorthand for the product x ⋅ x ⋅ x ⋅ x, that is, x

multiplied by itself four times. In the expression x^4 , x is called the base and 4 is

the power or exponent. We say “x raised to the fourth power” or simply “x to the

fourth.” Exponents have many useful properties.

Property 1 aanm=amn+

When multiplying two numbers whose bases are the same, we add their exponents.

EXAMPLES

23 · 2^4 = (2 · 2 · 2)(2 · 2 · 2 · 2) = 2^7

x^9. x^3 = x9+3 = x^12

Property 2 a

a

a

m

n

= mn−

(For the rest of the chapter, we will assume that a is not zero.)

When dividing two numbers whose bases are the same, we subtract their

exponents.

EXAMPLES

3

3

3333

33

3

4

2

11

11

==⋅⋅ ⋅^2

⋅

y

y

yy

7

3

==^73 −^4

Property 3 ()aanm= mn

If a number is raised to a power which is itself is raised to another power, we

multiply the exponents.

EXAMPLE

Rewrite using a single exponent.

(5^3 )^2 = (5 · 5 · 5)^2 = (5 · 5 · 5)(5 · 5 · 5) = 5^6

(x^6 )^7 = x(6)(7) = x^42

EXAMPLES

2

EXAMPLES

2

EXAMPLES EXAMPLES

EXAMPLE

Rewrite using a single exponent.

(5

EXAMPLE

Rewrite using a single exponent.