Powers: product, quotient, power of a power

The 3 fundamental rules

Operation	Rule
Product (same base)	$a^n \times a^m = a^{n+m}$
Quotient (same base)	$\dfrac{a^n}{a^m} = a^{n-m}$
Power of a power	$(a^n)^m = a^{n \times m}$

⚠️ These rules only apply when bases are identical.

Key special cases

• $a^0 = 1$ (for any $a \neq 0$)
• $a^1 = a$
• $a^{-n} = \dfrac{1}{a^n}$
• $a^{1/2} = \sqrt{a}$ (high school)

Rules for different bases

$(a \times b)^n = a^n \times b^n$

$\left( \dfrac{a}{b} \right)^n = \dfrac{a^n}{b^n}$

⚠️ There is no simple rule for $a^n + a^m$ or $(a + b)^n$.

Typical examples

$2^3 \times 2^5 = 2^{3+5} = 2^8 = 256$.

$\dfrac{10^7}{10^4} = 10^{7-4} = 10^3 = 1000$.

$(3^2)^4 = 3^{2 \times 4} = 3^8 = 6561$.

$5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25} = 0.04$.

Trick: quotient with negative exponents

$\dfrac{a^{-3}}{a^{-5}} = a^{-3 - (-5)} = a^{-3 + 5} = a^2$.

Common mistakes

1. $a^n + a^m = a^{n+m}$ (FALSE, that’s for the product).
2. $a^n \times b^n = (ab)^{2n}$ (false, it’s $(ab)^n$).
3. $(a + b)^2 = a^2 + b^2$ (false: $a^2 + 2ab + b^2$).
4. Forgetting that $a^0 = 1$.

?À toi de jouer

Simplify (2³)² × 2⁴?

A.2¹⁰ = 1024B.2⁹ = 512C.2¹² = 4096D.2⁶ = 64

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