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Exponentiation

Exponentiation

The use of exponents.

Key Formula

an=a×a×a×⋯×a⏟n factorsa^n = \underbrace{a \times a \times a \times \cdots \times a}_{n \text{ factors}}
Where:
  • aa = The base — the number being multiplied
  • nn = The exponent (or power) — how many times the base appears as a factor

Worked Example

Problem: Evaluate 3⁴ using exponentiation.
Step 1: Write the base (3) as a factor the number of times indicated by the exponent (4).
34=3×3×3×33^4 = 3 \times 3 \times 3 \times 3
Step 2: Multiply from left to right.
3×3=9,9×3=27,27×3=813 \times 3 = 9, \quad 9 \times 3 = 27, \quad 27 \times 3 = 81
Answer: 34=813^4 = 81

Why It Matters

Exponentiation provides a compact way to express repeated multiplication, just as multiplication is a shorthand for repeated addition. It appears throughout algebra, science, and finance — for example, compound interest formulas use exponentiation to model how money grows over time, and scientific notation uses powers of 10 to represent very large or very small numbers.

Common Mistakes

Mistake: Confusing exponentiation with multiplication: interpreting 343^4 as 3×4=123 \times 4 = 12.
Correction: 343^4 means 3 multiplied by itself 4 times, not 3 multiplied by 4. The correct result is 3×3×3×3=813 \times 3 \times 3 \times 3 = 81.

Related Terms

  • Exponent — The power to which a base is raised
  • Base (Number) — The number being raised to a power
  • Power — Another name for the result of exponentiation
  • Square Root — The inverse operation for exponent 2