grade 8 Laws of Exponents

Laws of Exponents

The laws of exponents are simple rules for multiplying, dividing, and raising powers to powers. Since two cubed times two squared is two multiplied five times, powers with the same base add their exponents. Division subtracts the exponents, and a power of a power multiplies them. Change the base and the exponent and each rule lines up.

Intuition — Repeated Multiplication

Base a2

Exponent n3

👀 What is a Power?

①2³ = 2 × 2 × 2 = 8 → multiply 2 three times

②Bigger base? Bigger exponent? The result explodes

③These rules of repeated multiplication are the laws of exponents

Law 1 — Same Base Product

Law 1: Multiplication

a^m × aⁿ = a^m+n

Same base, multiplied → add exponents

Laws 2 & 3 — Division and Power-of-Power

Law 2: Division

a^m ÷ aⁿ = a^m-n (m > n, a ≠ 0)

Same base, divided → subtract exponents. m=n → 1; m<n → a fraction. Sliders m·n are 1–4

Law 3: Power of a Power

(a^m)ⁿ = a^mn

Power raised to a power → multiply exponents

💡 Why Do Exponents Multiply?

①(a²)³ = a² × a² × a² → multiply a 2 times in 3 groups = 6 times

②(a^m)^n: doing m-times multiplication n times = m×n total

Laws 4 & 5 — Power of Products and Quotients

Power of a Product

(ab)ⁿ = aⁿ × bⁿ

Apply the exponent to each factor

Power of a Quotient

(ab)ⁿ = aⁿbⁿ

Apply the exponent to numerator and denominator

Work It Out

Example 1

Simplify a⁴ × a³.

Multiplying powers with the same base adds the exponents.

a⁴ × a³ = a⁴⁺³

Add the exponents.

= a⁷

▸ a⁷

Same base: multiply ⇒ add exponents, divide ⇒ subtract.

Example 2

Simplify (2a² b)³.

A power of a product raises each factor, including the coefficient.

= 2³ × (a²)³ × b³

Apply the laws of exponents.

= 8a⁶ b³

▸ 8a⁶ b³

Note (ab)ⁿ = aⁿ bⁿ and (a^m)ⁿ = a^mn.

Exam Wrap-up

Five Laws Summary

a^m·aⁿ=a^m+n, a^m÷aⁿ=a^m-n, (a^m)ⁿ=a^mn

(ab)ⁿ=aⁿ·bⁿ, (a/b)ⁿ=aⁿ/bⁿ

Grade-8 school exam type

If aⁿ ÷ a² = a⁵ (a ≠ 0), what is the natural number n?

①5

②6

③7

④8

⑤10

▸ ③ 7

Dividing powers with the same base subtracts the exponents.

aⁿ ÷ a² = aⁿ⁻² = a⁵

Compare the exponents.

n − 2 = 5 ⇒ n = 7

🎯 Exam Points

①Check whether bases match → if so, operate on exponents

②product→add, divide→subtract, power-of-power→multiply

③a⁰ = 1 (a ≠ 0). The exponent slider here is 1–6, so 0 is not reachable

④Mind signs: (-2)² = 4 but -2² = -4

⑤Parentheses change the result entirely

← Previous

Rational Numbers & Repeating Decimals

Polynomial Operations

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