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.
Slide the base and exponent to see why each rule holds.
Intuition — Repeated Multiplication
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Bar chart showing exponent growth as base and exponent change
2
3
👀 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
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Block visualization of a^m × a^n = a^(m+n)
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Law 1: Multiplication
am × an = am+n
Same base, multiplied → add exponents
Laws 2 & 3 — Division and Power-of-Power
Law 2: Division
am ÷ an = am-n (m > n, a ≠ 0)
Same base, divided → subtract exponents
Law 3: Power of a Power
(am)n = amn
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.
1
Multiplying powers with the same base adds the exponents.
a4 × a3 = a4+3
2
Add the exponents.
= a7
▸ a7
Same base: multiply ⇒ add exponents, divide ⇒ subtract.
Example 2
Simplify (2a2 b)3.
1
A power of a product raises each factor, including the coefficient.
= 23 × (a2)3 × b3
2
Apply the laws of exponents.
= 8a6 b3
▸ 8a6 b3
Note (ab)n = an bn and (am)n = amn.
Exam Wrap-up
Five Laws Summary
am·an=am+n, am÷an=am-n, (am)n=amn
(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
1
Dividing powers with the same base subtracts the exponents.
an ÷ a2 = an−2 = a5
2
Compare the exponents.
n − 2 = 5 ⇒ n = 7
🎯 Exam Points
①Check whether bases match → if so, operate on exponents