Multiplying Powers

Multiplying Powers 

Rule: For all numbers x and all integers m and n,
x^m • xⁿ = x^m+n
“This simply means when you are multiplying, and the bases are the same, you ADD the exponents.”
Consider:

Observe this rule at work in the following examples:

1. The bases are the same (all 5’s), so the exponents are added.
   5² × 5⁴ = 5²⁺⁴ = 5⁶
2. The bases are the same, so the exponents are added. Notice how the numbers in front of the bases (7 and 1) are being multiplied.
   7x³ • x⁵ = 7x⁸
3. The bases are the same (all a’s), so the exponents are added.
   2²(2³) (2⁵) = 2²⁺³⁺⁵ = 2¹⁰
4. The bases are the same (all x’s), so the exponents are added.
   Be careful when adding the negative exponent.
   5^-2 × 5⁶ = 5^-2+6 = 5⁴
5. The bases are the same, so the exponents are added.
   The numbers in front of the bases are multiplied.
   2x³ • 3x⁵ • 2x⁶ = 12x³⁺⁵⁺⁶ = 12x¹⁴

Take one more look at the distributive property at work with a set of parentheses, along with this new rule:
Use the distributive property to simplify:
3x² (2x³ + 4) = 3x² (2x³) + 3x² (4) = 6x⁵ + 12x²
The distributive property is applied in this problem. (Multiply each term inside the parentheses by the 3x² term.)
Then the exponents in the first portion are added since their bases are the same. The numbers in front (the coefficients) are multiplied.
Remember that you cannot add 6x⁵ and 12x² since they are not similar (like) terms.