Power laws with integer exponents

A power is shorthand for repeated multiplication:

a^3 = a · a · a
a^5 = a · a · a · a · a

If you want to multiply two powers of the same base, you do not have to

expand everything — there is a shortcut.

Rule 1 — product of powers

a^m · a^n = a^(m+n)

Think about why: `a^3 · a^2 = (a·a·a)·(a·a) = a^5`. The exponents add.

Common slip-up: it only works for the same base. `a^3 · b^2` does

not combine.

Rule 2 — quotient of powers

a^m ÷ a^n = a^(m−n),   when m ≥ n

a^5 ÷ a^2 = (a·a·a·a·a) ÷ (a·a) = a·a·a = a^3

For now we keep `m ≥ n` so the result is still a "regular" power. (At a

later level, `a^(−k) = 1 / a^k` gives meaning to negative exponents.)

Rule 3 — power of a power

(a^m)^n = a^(m·n)

(a^2)^3 = a^2 · a^2 · a^2 = a^6

The exponents multiply.

A short summary table

Worked example

(a^2)^3 · a^4 ÷ a^5
= a^6 · a^4 ÷ a^5    (power of power)
= a^10 ÷ a^5         (product)
= a^5                (quotient)

Try it yourself

• Power laws
• Simplify a linear expression
• Evaluate with 2 variables