## Positive Integer Exponents

### Meaning

3^{2} = 3 × 3

3^{3} = 3 × 3 × 3

In general *x ^{n}* =

*x*·

*x*·

*x*

· . . . ·

*x*(

*n*factors of

*x*)

*x* is the *base*, and *n* is the *exponent* (or *power*)

### Rules

**Product of Different Powers:** *a ^{m}a^{n}* =

^{ }

*a*

^{m + n}- IMPORTANT: all the numbers must have the same bases (the same ‘
*a*’)

**Example:** (4^{2})(4^{3}) = 4^{5}

This is easy to see if you write out the exponents:

(4^{2})(4^{3}) = (4 · 4) · (4 · 4 · 4) = 4 · 4 · 4 · 4 · 4 = 4^{5}

**WARNING**: Do not attempt to use this rule for addition:

4^{2} + 4^{3} is **NOT** 4^{5}. In fact there is no way to simplify *x ^{n}*

^{ + }

*x*if

^{m }*n*and

*m*are different powers.

**Power Raised to a Power:** _{}

**Example:** (4^{2})^{3} = 4^{6}

This is also easy to see if you expand the exponents:

(4^{2})^{3} = (4^{2})(4^{2})(4^{2}) = (4 · 4) (4 · 4) (4 · 4) = 4 · 4 · 4 · 4 · 4 · 4 = 4^{6}

There are more rules for combining numbers with exponents, but this is

enough for now.