Rules for computing with powers: the printable sheet

The 5 rules

Let $a, b \neq 0$ and $m, n$ integers.

Rule	Formula	Example
Product of powers (same base)	$a^m \times a^n = a^{m+n}$	$2^3 \times 2^5 = 2^8$
Quotient of powers (same base)	$\dfrac{a^m}{a^n} = a^{m-n}$	$\dfrac{3^7}{3^4} = 3^3$
Power of a power	$(a^m)^n = a^{m \times n}$	$(5^2)^3 = 5^6$
Product of different bases	$(a \times b)^n = a^n \times b^n$	$(2 \times 3)^4 = 2^4 \times 3^4$
Quotient of different bases	$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$	$\left(\dfrac{5}{2}\right)^3 = \dfrac{125}{8}$

Special cases

• $a^0 = 1$ (for any $a \neq 0$)
• $a^1 = a$
• $a^{-n} = \dfrac{1}{a^n}$ (negative exponent = reciprocal)

Traps

Trap 1 — Rules work on same base only. There’s no simple formula for $2^3 \times 3^3$ or $2^3 + 2^5$.

Trap 2 — $-3^2$ vs $(-3)^2$:

• $(-3)^2 = 9$ (square of negative is positive)
• $-3^2 = -9$ (compute $3^2 = 9$ first, then apply the −)

Examples

Simplify $\dfrac{5^7 \times 5^{-2}}{5^3}$:

• Numerator: $5^{7 + (-2)} = 5^5$
• Divided by $5^3$: $5^2 = 25$.

Simplify $\dfrac{(2^3)^4}{2^{10}}$:

• Numerator: $2^{12}$
• Divided by $2^{10}$: $2^2 = 4$.

?À toi de jouer

What is (2³ × 2²)²?

A.2¹⁰B.2⁷C.2⁶D.4¹⁰

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