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:    a^maⁿ = a^m + 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 x^n + x^m 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.