Powers: Introduction and Basics

A power is a short way to write repeated multiplication of the same number, such as 2⁵ = 2 · 2 · 2 · 2 · 2. In brief: in aⁿ, the base a is the number being multiplied and the exponent n tells how many times it appears as a factor. For example, 3⁴ = 3 · 3 · 3 · 3 = 81.

Powers appear in area and volume formulas, in compound interest, in computer memory sizes and in scientific notation. Before learning exponent laws or roots, it helps to be fluent with the basic meaning, the special cases and the order of operations. This guide covers all of that with worked examples and two reference tables.

What is a power, a base and an exponent?

A power consists of two parts. The base is the number that is multiplied, and the exponent is the small raised number that counts the factors. In 5³, the base is 5 and the exponent is 3, so 5³ = 5 · 5 · 5 = 125.

The whole expression is read “5 to the power of 3” or “5 to the third power”. The exponent 2 is usually read “squared” and the exponent 3 “cubed”, because x² gives the area of a square with side x and x³ gives the volume of a cube with edge x.

A power is not a multiplication of the base by the exponent. Compare 4³ = 4 · 4 · 4 = 64 with 4 · 3 = 12. This confusion is the most common beginner error, so it is worth checking each time.

What are the special cases of powers?

Three cases come up constantly and should be memorised.

• Exponent 1: a¹ = a, because the base appears as a factor only once.
• Exponent 0: a⁰ = 1 for every a other than 0. This follows from the exponent laws, for example 5³ ÷ 5³ = 1 and also 5³⁻³ = 5⁰.
• Base 1 and base 0: 1ⁿ = 1 for any n, and 0ⁿ = 0 for any positive n. The expression 0⁰ is treated differently in different contexts and is left undefined in basic algebra.

This article keeps to exponents that are whole numbers from 0 upward.

How do signs work with powers?

When the base is negative, the sign of the result depends on whether the exponent is even or odd. An even number of negative factors gives a positive product, and an odd number gives a negative product.

For example, (−2)⁴ = (−2)(−2)(−2)(−2) = 16, while (−2)³ = −8. Parentheses matter: (−2)⁴ = 16, but −2⁴ = −(2⁴) = −16, because the exponent applies only to the 2.

How do you calculate a power step by step?

1. Identify the base and the exponent. Check whether the sign belongs to the base, as in (−3)².
2. Write the base as a repeated factor the number of times given by the exponent.
3. Multiply from left to right, using partial products to keep the numbers small.
4. Check the sign using the even or odd rule.

Example: 6⁴

Write 6 · 6 · 6 · 6. Then 6 · 6 = 36, 36 · 6 = 216 and 216 · 6 = 1296. So 6⁴ = 1296.

Example: order of operations

Evaluate 3 + 2 · 4². Powers come before multiplication and addition. First 4² = 16, then 2 · 16 = 32, and finally 3 + 32 = 35.

Example: a fraction as a base

(2/3)³ = (2/3)(2/3)(2/3) = 8/27. The exponent applies to both numerator and denominator, so 2³ = 8 and 3³ = 27.

Squares and cubes table from 1 to 10

Powers of 2 and powers of 10 table

Powers of 10 simply add zeros. Powers of 2 grow quickly: each step doubles the previous value.

Where does exponent notation come from?

Ancient mathematicians wrote powers in words or with special names, such as “square” and “cube”, which come from geometry. The compact raised-exponent notation used today became standard in the seventeenth century, and René Descartes is commonly credited with popularising it in his work on geometry. Earlier writers used various abbreviations for unknowns and their powers, so the notation did not appear suddenly or from one person.

Where are powers used today?

Area uses squares and volume uses cubes, so units such as m² and m³ are powers of a length unit. In computing, memory sizes follow powers of 2: a byte has 2⁸ = 256 possible values. Compound interest, population growth and radioactive decay are modelled with exponents, and very large or very small measurements use powers of 10.

A quick example: a square garden with side 7 m has area 7² = 49 m², and a cube-shaped tank with edge 3 m holds 3³ = 27 m³.

Which mistakes are common with powers?

Multiplying base and exponent

2⁵ is 32, not 10. Write the repeated factors out whenever you are unsure.

Misplacing the sign

(−5)² = 25 but −5² = −25. Look for parentheses around the negative base.

Wrong order of operations

In 2 · 3² the exponent applies only to 3, giving 2 · 9 = 18, not 6² = 36.

Assuming a⁰ = 0

Any nonzero number to the power 0 equals 1. This is a consequence of the exponent laws.