rule
For all numbers x and all integers m and n,
rule1
rule1pic

When you multiply,
and the bases are the same,
you ADD the exponents.

When in doubt, expand the terms (as shown at the right) to see what is happening.

rule1m

rule1m2

4444

Examples:

1.  32 × 34 = 32+4 = 36
The bases are the same (both 3's), so the exponents are added.
2.  22(23) (25) = 22+3+5 = 210
The bases are the same (all 2's), so the exponents are added.

3.  x3x5x6 = x3+5+6 = x14
The bases are the same (all x's), so the exponents are added.

4.  32 + 34 ≠ 32+4
Oops!! This problem is NOT multiplication. This rule does not apply to addition.

5.  (-2)3 • 125 • (-2)6 • 12 = (-2)9 • 126
Be sure to only add the exponents for the bases that are the same. Also, do not forget that 12 has an exponent of 1 that is implied, but not written.

6.  92 × 24 = ?
This multiplication rule cannot be applied to this problem since the bases are not the same. In some such cases, as you will see in Example 7, it may be possible to rewrite one of the terms in the same base as the other term, but that is not the case here,

7.  83 × 25 = 29 × 25 = 214
Sneaky one!!! In this problem 8 can be written as 2 cubed, thus creating the same base for both terms.. 8888. You get to apply the Rule twice in this one problem.

8.  64 × 36 = 64 × 62 = 66
The 36 can be written as a power of 6 making the bases the same value. The needed CHANGE was easier to see in this problem, than in Example 7.
9.  5a2 • 2a3a4 = 5 • 2 • 1 • a2+3+4
      = 10a9
The bases are the same (all a's), so the exponents are added. Notice how the numbers in front of the bases (5, 2, and 1) are being multiplied.
10.  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.

 

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