Algebra Honors (Period 6 & 7)

Negative Exponents 7-9
If a is a nonzero real number and n is a positive integer
a^-n = 1/aⁿ
so
10^-3 = 1/10³ = 1/1000
5^-4= 1/5⁴ = 1/625

16^-1 = 1/16

The rule of exponents for division (page 190) will help you understand why
a^-n = 1/aⁿ
recall that for m > n a^m/aⁿ= a^m-n
For example
a⁷/a³ = a^7-3= a⁴
you can also apply this rule when m < n that is when m - n becomes a negative number. For example a³/a⁷ = a^3-7 = a^-4
since
a⁷/a³ and a³/a⁷ are reciprocals then
a⁴ and a^-4 must also be reciprocals.
Thus
a^-4= 1/a⁴
a⁵/ a⁵ = a^5-5 = a⁰
But you already know that a⁵/a⁵ = 1
SO, definition of a⁰
a⁰ = 1
However, the expression 0⁰ has no meaning

All the rules for positive exponents also hold for zero and negative exponents.

Summary of Rules for Exponents
Let m and n be any integers
Let a and b be any non zero integers
Review—>But you should really know these because of our Powers Project
Products of Powers
b^mbⁿ = b^m+n
Example with negative exponents
2³⋅2^-5 = 2^3+(-5) = 2^-2 = 1/2² = 1/4
Quotient of Powers
b^m ÷ bⁿ = b^m-n
Example with negative exponents
6³÷6⁷= 6^3-7= 6^-4= 1/6⁴= 1/1296
Power of Powers
(b^m)ⁿ = b^mn
Example with negative exponents
(2³)^-2 = 2^-6 = 1/2⁶ = 1/64
Power of a Product
(ab)^m= a^mb^m
Example with negative exponents
(3x)^-2 = 3^-2 ⋅x^-2 = 1/3²⋅1/x² = 1/9x²
Power of a Quotient
(a/b)^m= a^m/b^m
Example with negative exponents
(3/5)^-2= 3^-2/5^-2= (1/3²)/ (1/5²)= 1/3² ÷ 1/5² which means
1/3² ⋅5²/1= 5²/3²= 25/9