Exponents & Powers
An exponent represents repeated multiplication, where a base number is multiplied by itself a specified number of times. The expression 2³ means 2 × 2 × 2, which equals 8. Exponents follow specific rules that make calculations with large numbers more manageable, such as 2⁴ × 2³ = 2⁷ = 128.
Why it matters
Exponents appear throughout science and finance, from calculating compound interest to measuring exponential growth in populations. In computer science, powers of 2 determine storage capacity — 2¹⁰ = 1,024 bytes in a kilobyte. Scientists use scientific notation with exponents to express massive numbers like 6.022 × 10²³ (Avogadro's number) or tiny measurements like 1.6 × 10⁻¹⁹ (charge of an electron). Population growth models use exponential functions where a city growing at 3% annually reaches 1.03ⁿ times its original size after n years. In algebra courses following CCSS 8.EE standards, exponent rules become essential for polynomial operations, logarithms, and advanced functions in calculus.
How to solve exponents & powers
Exponents & Powers
- am × an = am+n — same base, add exponents.
- am ÷ an = am−n — same base, subtract.
- (am)n = am×n — power of power, multiply.
- a0 = 1, a-n = 1/an.
Example: 2³ × 2⁴ = 2⁷ = 128.
Worked examples
43 = _______
Answer: 64
- Multiply 4 by itself 3 times → 4 × 4 × 4 = 64 — 4^3 means 4 multiplied 3 times.
42 × 42 = _______
Answer: 44
- Same base → add exponents → 4(2+2) = 44 — When multiplying same base, add the powers.
24 = _______
Answer: 16
- Evaluate → 2 × 2 × 2 × 2 = 16 — Multiply repeatedly.
Common mistakes
- Adding exponents instead of multiplying the base: writing 2³ + 2³ = 2⁶ instead of 2³ + 2³ = 8 + 8 = 16
- Applying exponent rules to different bases: calculating 3² × 4² as (3 × 4)⁴ = 12⁴ instead of 9 × 16 = 144
- Confusing negative exponents with negative results: computing 2⁻³ as -8 instead of 1/8 or 0.125