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§ Algebra

Exponents & Powers

§ Algebra

Exponents & Powers

CCSS.8.EECCSS.HSA.SSE3 min read

An exponent represents how many times a base number multiplies by itself. In the expression 3⁴, the base is 3 and the exponent is 4, meaning 3 × 3 × 3 × 3 = 81. Exponents follow specific rules that make calculations with large numbers more manageable.

§ 01

Why it matters

Exponents appear throughout GCSE mathematics, from calculating compound interest rates to understanding scientific notation in physics. A £1000 investment growing at 5% annually becomes £1000 × 1.05⁵ = £1276.28 after 5 years. In science, the speed of light (300,000,000 metres per second) becomes 3 × 10⁸ m/s using exponential notation. Computer storage uses powers of 2: a gigabyte contains 2³⁰ bytes, approximately 1 billion. Surds like √8 = 2√2 connect to fractional exponents where √8 = 812. Students encounter exponential growth in population studies, radioactive decay in physics, and area calculations where a square with side length n has area n². These foundations support A-level calculus and university mathematics.

§ 02

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.

§ 03

Worked examples

Beginner§ 01

What do you get when you multiply 2 by itself 2 times?

Answer: 4

  1. Multiply 2 by itself 2 times → 2 × 2 = 4 — 2^2 means 2 multiplied 2 times.
Easy§ 02

64 = _______

Answer: 1296

  1. Evaluate → 6 × 6 × 6 × 6 = 1296 — Multiply repeatedly.
Medium§ 03

92 = _______

Answer: 81

  1. Evaluate → 9 × 9 = 81 — Multiply repeatedly.
§ 04

Common mistakes

  • Adding exponents instead of multiplying the base: writing 2³ = 6 instead of 8
  • Applying the power rule incorrectly: writing (3²)³ = 3⁵ instead of 3⁶
  • Confusing negative exponents with negative answers: thinking 2⁻³ = -8 instead of 1/8
§ 05

Frequently asked questions

What does a zero exponent mean?
Any non-zero number raised to the power of 0 equals 1. For example, 5⁰ = 1 and 100⁰ = 1. This follows from the division rule: 5³ ÷ 5³ = 5⁰ = 1, since any number divided by itself equals 1.
How do negative exponents work?
A negative exponent means taking the reciprocal. 2⁻³ = 1/2³ = 1/8. This extends the pattern: as exponents decrease (2³, 2², 2¹, 2⁰), each result halves. Continuing this pattern gives 2⁻¹ = 1/2 and 2⁻² = 1/4.
What is the difference between 2³ and 3²?
The base and exponent positions matter completely. 2³ = 2 × 2 × 2 = 8, whilst 3² = 3 × 3 = 9. The first number is the base (what gets multiplied), and the superscript is the exponent (how many times to multiply).
Can you multiply powers with different bases?
Powers with different bases cannot be simplified using exponent rules. 2³ × 3² stays as 8 × 9 = 72. The multiplication and division rules (a^m × a^n = a^(m+n)) only work when the bases are identical.
How do you calculate fractional exponents?
Fractional exponents represent roots. 8^(1/3) means the cube root of 8, which equals 2. More generally, a^(m/n) means the nth root of a^m. For example, 16^(3/4) = (⁴√16)³ = 2³ = 8.
§ 06

See also

BaseExpressionAreaMeanRoot
§ 06

Where to next?

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