1. Simplifying Expressions with Integral Exponents

Laws of Exponents

Integral Exponents

Integral here means "integer". So the exponent (or power) is an integer. [That is, either a negative whole number, 0, or a positive whole number.]

Definition: a^m means "multiply a by itself m times". That is:

a^m = a × a × a × a × a × ... × a
[We do the multiplication m times.]

Note: This definition only really holds for m > 0, since it doesn't make a lot of sense if m is negative. (You can't multiply something by itself -3 times! And how does multiplying something by itself 0 times give 1?)

In such cases we have to rely on patterns and conventions to define what is going on. See below for zero and negative exponents.

Examples

(1) y⁵ = y × y × y × y × y

(2) 2⁴ = 2 × 2 × 2 × 2 = 16.

Multiplying Expressions with the Same Base

Definition: a^m × aⁿ = a^m+n

Let's see how this works with an example.

Example

b⁵ × b³

= (b × b × b × b × b) × (b × b × b)

= b⁸ (that is, b⁵⁺³)

Dividing Expressions with the Same Base

Definition: (Of course, a ≠ 0)

It may be easiest to see how this one works with an example.

Example

We cancel 2 of the b's from the numerator and the two b's from the denominator of the fraction. The result is equivalent to b^{7 − 2}.

Repeated Multiplication of a Number Raised to a Power

(a^m)ⁿ	= (a^m) × (a^m) × (a^m) × ... × (a^m) [We multiply n times].
	= a^mⁿ

So we write:

Definition: (a^m)ⁿ = a^mn

Example

(p³)²	= p³ × p³
	= (p × p × p) × (p × p × p)
	= p⁶

A Product Raised to an Integral Power

Definition: (ab)ⁿ = aⁿbⁿ

Example

(5q)³ = 5³q³ = 125q³

A Fraction Raised to an Integral Power

Definition:

Example

integral exponents

Zero Exponents

Definition: a⁰ = 1 (a ≠ 0)

Example

7⁰ = 1

Note: a⁰ = 1 is a convention, that is, we agree that raising any number to the power 0 is 1. We cannot multiply a number by itself zero times.

In the case of zero raised to the power 0 (0⁰), mathematicians have been debating this for hundreds of years. It is most commonly regarded as having value 1, but is not so in all places where it occurs. That's why we write a ≠ 0.