Exponents and Roots
The expression 4x is shorthand for x + x + x + x, that is x added to itself four times. Likewise x4 is shorthand for x · x · x · x—x multiplied by itself four times. In x4, x is called the base and 4 is the power or exponent. We say ‘‘x to the fourth power’’ or simply ‘‘x to the fourth.’’ There are many useful exponent properties. For the rest of the chapter, a is a nonzero number.
Property 1 anam
= am+n
When
multiplying two powers whose bases are the same, add the exponents.
Examples
23
· 24 = (2 · 2 · 2)(2 · 2 · 2 · 2) = 27
x9
· x3 = x12
Property 2 
When
dividing two powers whose bases are the same, subtract the denominator’s power
from the numerator’s power.
Examples
Property 3 (an)m = anm
If you have
a quantity raised to a power then raised to another power, multiply the
exponents.
Examples
(53)2
= (5 · 5 · 5) (5 · 5 · 5) = 56
(x6)7 = x(6)(7) = x42
Be careful,
Properties 1 and 3 are easily confused.
Property 4 a0 = 1
Any nonzero
number raised to the zero power is one. We will see that this is true by
Property 2 and the fact that any nonzero number over itself is one.
From
this we can see that 40 must be 1.
Practice
Rewrite
using a single exponent.
Solutions
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