Grades Grade 8 Expressions & EquationsProperties of Integer Exponents

Properties of Integer Exponents

✏️ Expressions & Equations

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Properties of Integer Exponents

Exponent rules let you simplify any expression with integer powers — including negative and zero exponents.

When multiplying or dividing powers with the same base, you add or subtract the exponents. Negative exponents represent reciprocals, and any nonzero base raised to the power of 0 equals 1.

a^m \cdot a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n}

(a^m)^n = a^{mn} \qquad a^{-n} = \frac{1}{a^n} \qquad a^0 = 1 \; (a \neq 0)

🧮Simplifying with exponent rules

Simplify: (2³ × 2⁻¹) ÷ 2²

Step 1: 2³ × 2⁻¹ = 2^(3+(−1)) = 2²
Step 2: 2² ÷ 2² = 2^(2−2) = 2⁰ = 1

\frac{2^3 \cdot 2^{-1}}{2^2} = \frac{2^{3+(-1)}}{2^2} = \frac{2^2}{2^2} = 2^0 = 1

Exponent rules at a glance:

1Product rule: aᵐ · aⁿ = aᵐ⁺ⁿ (add exponents, same base)

2Quotient rule: aᵐ / aⁿ = aᵐ⁻ⁿ (subtract exponents, same base)

3Power rule: (aᵐ)ⁿ = aᵐⁿ (multiply exponents)

4Zero exponent: a⁰ = 1 for any a ≠ 0

5Negative exponent: a⁻ⁿ = 1/aⁿ (flip to denominator)

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Remember This!

A negative exponent does NOT make the base negative — it moves the factor to the denominator. 3⁻² = 1/9, not −9.

✏️ Try It!

Simplify: x⁵ · x⁻³. Which expression is equivalent?

Practice with CalcVerse

⚡Exponent Rules

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