Exponents and Powers

A shorthand for repeated multiplication

Multiplication is a shortcut for repeated addition: 4 × 3 means "add 4 three times." Exponents take the next step — they are a shortcut for repeated multiplication.

Instead of writing 2 × 2 × 2 × 2 × 2, we write 2⁵. That little raised number saves a lot of space, and as you will see, it makes huge numbers easy to handle. If you are still building confidence with multiplication, the Introduction to Multiplication lesson is a good place to start.

Base and exponent

Every power has two parts:

base^exponent

In 2⁵:

• the 2 is the base — the number being multiplied.
• the 5 is the exponent (or power, or index) — how many times the base is used as a factor.

So 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. We read it as "two to the power of five" or "two to the fifth."

A common mistake is to multiply the base by the exponent. 2⁵ is not 2 × 5 = 10. The exponent counts factors; it does not multiply.

Squares and cubes

Two powers are so common they have special names.

• Squaring is raising to the power 2: 5² = 5 × 5 = 25, read "five squared." The name comes from the area of a square with side 5 — see Area and Perimeter.
• Cubing is raising to the power 3: 4³ = 4 × 4 × 4 = 64, read "four cubed." It comes from the volume of a cube with side 4.

Power	Meaning	Value
3²	3 × 3	9
4²	4 × 4	16
2³	2 × 2 × 2	8
5³	5 × 5 × 5	125
10⁴	10 × 10 × 10 × 10	10 000

Notice the last row: powers of 10 are how we write large numbers. 10⁴ is 1 followed by four zeros. This is the engine behind scientific notation.

The laws of exponents

When powers share the same base, three handy rules let you simplify without expanding everything. Understanding why each one works is more useful than memorising it.

1. Multiplying — add the exponents

$$ a^m \times a^n = a^{m+n} $$

Why: 2³ × 2⁴ = (2×2×2) × (2×2×2×2). Count the 2s: there are 3 + 4 = 7 of them, so the answer is 2⁷.

2. Dividing — subtract the exponents

$$ a^m \div a^n = a^{m-n} $$

Why: 2⁵ ÷ 2² = (2×2×2×2×2) ÷ (2×2). Two of the 2s on top cancel with the two on the bottom, leaving 5 − 2 = 3 of them, so 2³.

3. Power of a power — multiply the exponents

$$ (a^m)^n = a^{m \times n} $$

Why: (2³)² means 2³ × 2³. That is two lots of three 2s, which is 6 of them in all: 2⁶.

Rule	Example	Result
Multiply: add	5² × 5³	5⁵
Divide: subtract	7⁶ ÷ 7²	7⁴
Power of power: multiply	(3²)⁴	3⁸

Warning: these rules only work when the bases are the same. You cannot simplify 2³ × 5² with them, because 2 and 5 are different bases.

The power of zero

What is 2⁰? Look at a pattern of dividing by the base each time the exponent drops by one:

• 2³ = 8
• 2² = 8 ÷ 2 = 4
• 2¹ = 4 ÷ 2 = 2
• 2⁰ = 2 ÷ 2 = 1

The pattern forces 2⁰ = 1, and the same argument works for any non-zero base. So anything (except 0) to the power 0 equals 1. It also drops straight out of the division rule: a³ ÷ a³ = a⁰, and any number divided by itself is 1.

Negative exponents

Continue the same halving pattern below zero:

• 2⁰ = 1
• 2⁻¹ = 1 ÷ 2 = 1/2
• 2⁻² = (1/2) ÷ 2 = 1/4

So a negative exponent means a reciprocal — one over the positive power:

$$ a^{-n} = \frac{1}{a^{n}} $$

Example: 5⁻² = 1/5² = 1/25. A negative exponent does not make the answer negative; it makes it a fraction.

Worked example — combining rules

Simplify (2³ × 2⁴) ÷ 2⁵.

1. Multiply on top (add): 2³ × 2⁴ = 2⁷.
2. Divide (subtract): 2⁷ ÷ 2⁵ = 2² = 4.

Practice activity

Work these out, then check each one.

1. Evaluate 2⁴, 3³, and 10³.
2. Simplify 6² × 6³ as a single power.
3. Simplify 8⁷ ÷ 8⁴ as a single power.
4. Find the value of 9⁰ + 1.
5. Write 4⁻² as a fraction.

Answers: 1) 16, 27, 1000 2) 6⁵ 3) 8³ 4) 1 + 1 = 2 5) 1/16.

Why this matters

Exponents are the language of growth and scale. They power scientific notation (writing the distance to the Sun without 11 zeros), compound interest, computer memory measured in powers of 2, and the area and volume formulas in geometry. The whole topic rests on one definition — repeated multiplication — and three logical rules that you can rebuild any time by counting the factors. Understand the why, and exponents become a tool you trust rather than a set of rules to memorise.