Multiplying and dividing powers: the rules — Mathematics, 11–13
Same base: add or subtract the exponents. Same exponent: multiply or divide the bases. And why counting factors proves it.
Four rules: same base, same exponent
Same base: when you multiply, the exponents add; when you divide, they subtract (the larger exponent first). 2³ × 2⁴ = 2⁷ and 2⁷ : 2³ = 2⁴. Same exponent: the exponent stays and the bases are multiplied or divided. 2³ × 5³ = (2 × 5)³ = 10³, and 12³ : 4³ = (12 : 4)³ = 3³, because 12 : 4 is exact. The base or the exponent must match: with neither, there is no rule.
Why the rules work: count the factors
A power is a list of equal factors, so just count them. 2³ × 2⁴ = (2 × 2 × 2) × (2 × 2 × 2 × 2): 3 + 4 = 7 twos, so 2⁷. In 2⁷ : 2³, three of the seven 2s cancel out and 7 − 3 = 4 are left: 2⁴. With the same exponent the order of the factors does not matter: 2³ × 5³ = (2 × 2 × 2) × (5 × 5 × 5) = (2 × 5) × (2 × 5) × (2 × 5) = 10³. Nothing here depends on 2 or 5, so the rules hold for any natural bases.
Four calculations, one rule each
(a) Same base, multiply: 2⁵ × 2³ = 2⁵⁺³ = 2⁸ = 256. (b) Same base, divide: 5⁹ : 5⁶ = 5⁹⁻⁶ = 5³ = 125. (c) Same exponent, multiply: 2⁴ × 5⁴ = (2 × 5)⁴ = 10⁴ = 10 000. (d) Same exponent, divide: 12³ : 4³ = (12 : 4)³ = 3³ = 27. Check (d) without the rule: 12³ = 1728, 4³ = 64 and 1728 : 64 = 27. The rule saved you from a big number.
The trap: changing the base, or using the rule where it does not apply
A common slip: 2³ × 2⁴ is not 4⁷ (the base stays, it is not multiplied) and not 2¹² (exponents add, they are not multiplied). It is 2⁷ = 128, while 4⁷ = 16 384. Another slip is using the rules everywhere. 2³ × 3⁴ has different bases and different exponents, so no rule simplifies it: calculate 8 × 81 = 648. And 2³ + 2⁴ = 8 + 16 = 24 is not 2⁷: these rules are only for multiplying and dividing.
Where you see it outside school: big numbers made easy
With powers of 10 the rule saves zeros: 1000 × 10 000 = 10³ × 10⁴ = 10⁷ = 10 000 000, and 10⁸ : 10⁵ = 10³ = 1000. It also helps mental calculation: 25² × 4² = (25 × 4)² = 100² = 10 000, much easier than 625 × 16. In a game where the points double every round, 10 rounds multiply them by 2¹⁰ = 1024; another 10 rounds multiply again, and 2¹⁰ × 2¹⁰ = 2²⁰ = 1 048 576 altogether.
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