Positive Integer Exponents

Meaning

32 = 3 × 3

33 = 3 × 3 × 3

In general xn = x · x · x
· . . . · x (n factors of x)

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

Rules

Product of Different Powers:    aman = am + n

  • IMPORTANT: all the numbers must have the same bases  (the same ‘a’)

Example:  (42)(43) = 45

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

(42)(43) = (4 · 4) · (4 · 4 · 4) = 4 · 4 · 4 · 4 · 4 = 45

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

42 + 43 is NOT 45. In fact there is no way to simplify xn + xm if n and m are different powers.

Power Raised to a Power:   

Example:  (42)3 = 46

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

(42)3 = (42)(42)(42) = (4 · 4) (4 · 4) (4 · 4) = 4 · 4 · 4 · 4 · 4 · 4 = 46

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