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The Laws of Exponents: Product, Quotient & Power Rules

Three rules do almost all the work. Each one is just bookkeeping on factors — once you see why they’re true, you’ll never have to memorize them. We add exponents to multiply, subtract to divide, and multiply to raise a power to a power.

Multiply: add the exponents

The product rule says that to multiply two powers with the same base, you keep the base and add the exponents: b^m · b^n = b^(m+n). Why? Because each power is just a pile of factors, and putting two piles together gives one bigger pile. Count and you’ll see it.

2^3 · 2^4 = (2·2·2) · (2·2·2·2)
          = 2·2·2·2·2·2·2
          = 2^7        (3 + 4 = 7)

Check:  2^3 · 2^4 = 8 · 16 = 128 = 2^7  ✓
The product rule is just counting factors: 3 of them plus 4 of them makes 7.

Divide: subtract the exponents

The quotient rule is the mirror image: b^m / b^n = b^(m−n) for the same nonzero base. Dividing cancels matching factors from top and bottom, so what survives is the difference of the counts.

x^5 / x^2 = (x·x·x·x·x) / (x·x)
          = x·x·x          (two factors cancel)
          = x^3            (5 − 2 = 3)

With coefficients:  20y^7 / 5y^3 = (20/5)·y^(7−3) = 4y^4
Cancel the shared factors; the leftover count is m − n.

Power of a power, product, and quotient

The power rule handles a power raised to another power: (b^m)^n = b^(m·n). You have n copies of b^m, each carrying m factors, so altogether m·n factors. Two close cousins also live here: a product to a power spreads out, (ab)^n = a^n b^n, and so does a quotient, (a/b)^n = a^n / b^n.

(x^3)^4 = x^3 · x^3 · x^3 · x^3 = x^12   (3 · 4 = 12)
(2x)^3  = 2^3 · x^3 = 8x^3
(x/3)^2 = x^2 / 3^2 = x^2 / 9

Mixed:  (2x^2 y)^3 = 2^3 · x^6 · y^3 = 8x^6 y^3
Power to a power multiplies; product and quotient powers distribute to every factor.