What is an exponent?
In aⁿ, a is the base and n is the exponent (also called index or power). It means a multiplied by itself n times.
Example: 5³ = 5 × 5 × 5 = 125. Read it as “5 to the power 3” or “5 cubed”.
Watch the sign: (−3)² = 9 but −3² = −9, because the exponent belongs only to the 3 when there are no brackets.
The main laws (same base)
- Product: aᵐ × aⁿ = aᵐ⁺ⁿ (count all the factors together).
- Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ, a ≠ 0 (matching factors cancel).
- Power of a power: (aᵐ)ⁿ = aᵐⁿ (n groups of m factors).
- Power of a product: (ab)ⁿ = aⁿbⁿ.
- Power of a quotient: (a/b)ⁿ = aⁿ/bⁿ, b ≠ 0.
The laws work only when the bases are the same (or the exponents are the same, for the last two). 2³ × 3² cannot be joined into one power.
Zero and negative (integer) exponents
Use the quotient law on a⁵ ÷ a⁵. The answer must be 1 (anything divided by itself), and the law gives a⁰. So a⁰ = 1 for a ≠ 0.
Now a² ÷ a⁵ = a⁻³, but cancelling gives 1/a³. So a⁻ⁿ = 1/aⁿ. A negative exponent does not make the number negative: 2⁻³ = 1/8.
A handy trick: (a/b)⁻ⁿ = (b/a)ⁿ. Flip the fraction and make the exponent positive.
Standard form uses this: 0.00045 = 4.5 × 10⁻⁴.
Rational (fractional) and real exponents
(a^(1/2))² = a^1 = a, so a^(1/2) must be √a. In general a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ for a > 0.
Example: 8^(2/3) = (∛8)² = 2² = 4. Take the root first: the numbers stay small.
All the laws still hold for rational exponents. For irrational exponents such as 2^√2, we get closer and closer using 2^1.4, 2^1.41, 2^1.414..., so the laws extend to all real exponents (for a positive base).
Simple exponential equations
If two powers of the same base are equal, their exponents are equal: if aˣ = aʸ (a > 0, a ≠ 1) then x = y.
Example: 3ˣ⁺¹ = 81 = 3⁴, so x + 1 = 4 and x = 3.
Example: 4ˣ = 32. Write both as powers of 2: 2²ˣ = 2⁵, so 2x = 5 and x = 2.5.
Real-world: bacteria that double every hour: N = 100 × 2ᵗ. When is N = 3200? 2ᵗ = 32 = 2⁵, so t = 5 hours.
Key formulas and definitions
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ
- (aᵐ)ⁿ = aᵐⁿ
- (ab)ⁿ = aⁿbⁿ
- (a/b)ⁿ = aⁿ/bⁿ
- a⁰ = 1 (a ≠ 0)
- a⁻ⁿ = 1/aⁿ
- a^(m/n) = (ⁿ√a)ᵐ
Worked examples
1. Simplify 3⁴ × 3².
Same base, multiply → add: 3⁴⁺² = 3⁶ = 729.
2. Simplify 7⁸ ÷ 7⁵.
Same base, divide → subtract: 7⁸⁻⁵ = 7³ = 343.
3. Find the value of (2³)² × 2⁻⁴.
(2³)² = 2⁶. Then 2⁶ × 2⁻⁴ = 2² = 4.
4. Evaluate (2/5)⁻².
Flip and make positive: (5/2)² = 25/4 = 6.25.
5. Evaluate 27^(2/3) + 16^(−1/2).
27^(2/3) = (∛27)² = 3² = 9. 16^(−1/2) = 1/√16 = 1/4. Total = 9.25.
6. Simplify (x³y⁻²)² ÷ (x²y)⁻¹ and write with positive exponents.
(x³y⁻²)² = x⁶y⁻⁴. Dividing by (x²y)⁻¹ is multiplying by x²y. So x⁸y⁻³ = x⁸/y³.
7. Solve 2ˣ × 4 = 128.
4 = 2², 128 = 2⁷. So 2ˣ⁺² = 2⁷, x + 2 = 7, x = 5.
Common mistakes
- Multiplying the exponents in a product: 2³ × 2² is 2⁵, not 2⁶.
- Multiplying the bases: 3² × 3⁴ is 3⁶, not 9⁶.
- Thinking a negative exponent makes the answer negative: 5⁻² = 1/25, a positive number.
- Thinking a⁰ = 0: any non-zero number to the power 0 is 1.