What is a Power (Exponentiation)?

A power (or exponent) represents repeated multiplication. b^n means multiplying b by itself n times.

b^n = b × b × b × ... (n times)
Example: 2^8 = 2×2×2×2×2×2×2×2 = 256

Power Rules

Any number to the power of 0 = 1 (e.g., 5⁰ = 1)
Any number to the power of 1 = itself (e.g., 7¹ = 7)
Negative exponent = 1 / (positive power) (e.g., 2⁻³ = 1/8)
Fractional exponent = root (e.g., 9^0.5 = √9 = 3)

Powers Reference Table

Powers appear everywhere in mathematics, science, and engineering. Here are the most commonly needed values:

Base	²	³	⁴	⁵
2	4	8	16	32
3	9	27	81	243
4	16	64	256	1,024
5	25	125	625	3,125
6	36	216	1,296	7,776
10	100	1,000	10,000	100,000

Special Power Cases

Any base to power 0: b⁰ = 1 (e.g., 7⁰ = 1)
Any base to power 1: b¹ = b (e.g., 9¹ = 9)
0 to any positive power: 0ⁿ = 0 (n > 0)
Negative base, even power: (−3)⁴ = 81 (positive)
Negative base, odd power: (−3)³ = −27 (negative)

Worked Examples

Example 1 — Area of a Square

A square has side 12 cm. Area = 12² = 144 cm².

Example 2 — Volume of a Cube

A cube has side 5 m. Volume = 5³ = 125 m³.

Example 3 — Compound Interest

$1,000 invested at 10% for 5 years: A = 1000 × 1.1⁵ = 1000 × 1.61051 = $1,610.51

1.1⁵ = 1.1 × 1.1 × 1.1 × 1.1 × 1.1 = 1.61051

Real-World Power Calculations

Finance: Compound interest A = P(1+r)ⁿ — the exponent is compounding periods
Physics: Kinetic energy KE = ½mv² — velocity is squared
Geometry: Area formulas use squares (r², s², bh); volume uses cubes (r³, s³)
Signal processing: Power of a signal is proportional to amplitude squared
Earthquake magnitude: Each unit on the Richter scale = 10× increase in amplitude

Power Calculator (Base & Exponent)