Numbers & Functions · 5 Exponentials and Logarithms

5.3 The laws of exponents (and why every one is true)

Let’s nail down the rules as a toolkit. There are seven, but here’s the good news: they’re not seven separate facts. There are really just a few, and the rest are forced.

♦Why it’s true — skip on a first read, but the real brain-training Each law below comes in two parts: the law itself (your toolkit) and a Why (the proof). On a first read you can take the bold laws and skip every Why — you’ll be able to do all the problems with just the laws. But the proofs all come down to one homely idea — an exponent just counts how many copies you’re multiplying — so they’re satisfying brain-training. Law 1 — Product rule: am · an = am+n. Why: am is m copies of a; an is n more. Side by side, that’s m + n copies. Multiplying powers adds the exponents, because you’re pooling two piles into one. This is the master rule — the other six all flow from it.

Law 2 — Quotient rule: am an = am−n. Why: dividing cancels copies.

Each bottom copy cancels a top copy, leaving m −n. The mirror of Law 1. Law 3 — Power of a power: (am)n = amn. Why: n copies of am, each m copies of a, is m × n copies.

Law 4 — Power of a product: (ab)n = anbn. Why: sort the n copies of ab into a pile of a’s and a pile of b’s.

The next three are forced consequences of Law 1, not new rules: Law 5 — Zero exponent: a0 = 1. Why: am · a0 = am, and only 1 does nothing under multiplication.

Law 6 — Negative exponent: a−n = 1 an . Why: an · a−n = a0 = 1, so a−n is the reciprocal.

Law 7 — Fractional exponent: am/n = n√ am. Why: (a1/n)n = a1 = a, so a1/n is the nth root.

The entire toolkit is really one idea — exponents count copies — plus its forced consequences. If you ever blank on Law 6 or 7 in an exam, rebuild it from Law 1. The method outlasts the memory.

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