A power a^n is shorthand for multiplying a by itself n times. The number a is the base and n is the exponent. For example 2^5 = 2 · 2 · 2 · 2 · 2 = 32.

Why is any number to the power of zero equal to 1?

Because dividing powers subtracts exponents: a^n / a^n = a^(n−n) = a^0, and at the same time any number divided by itself is 1. For the law to keep holding, a^0 must be 1 (for a ≠ 0).

What does a negative exponent mean?

A negative exponent means a reciprocal: a^(−n) = 1 / a^n. For example 2^(−3) = 1/2^3 = 1/8. It does not mean a negative result — 2^(−3) is a positive number.

What is the difference between (−2)^4 and −2^4?

The bracket decides what the base is. In (−2)^4 the base is −2, so the result is 16. In −2^4 only the two is raised and the minus stays in front, so the result is −16.

Can powers with different bases be added by adding the exponents?

No. The laws of exponents apply to MULTIPLYING and DIVIDING powers with the same base. The sum 2^3 + 2^4 cannot be written as 2^7 — you have to compute 8 + 16 = 24.

What is scientific notation for?

Scientific notation writes a number as a · 10^n with 1 ≤ a < 10. It keeps very large and very small quantities short: 300,000,000 is 3 · 10^8, and 0.00042 is 4.2 · 10^(−4).

Intermediate level

Powers

A power is shorthand for multiplying the same factor by itself. Learn the base and the exponent, the laws of exponents, zero and negative exponents, and scientific notation.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

MultiplicationMultiplication is repeated addition of the same number. Learn the names of the factors and the product, the laws of multiplication (commutativity, associativity, distributivity), the times tables and column multiplication.

All formulas

Definition of a power
$a^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ factors}}$
a is the base, n is the exponent
Multiplying powers with the same base
$a^m \cdot a^n = a^{m+n}$
add the exponents
Dividing powers with the same base
$\frac{a^m}{a^n} = a^{m-n}$
subtract the exponents
A power of a power
$(a^m)^n = a^{m \cdot n}$
multiply the exponents
A power of a product
$(a \cdot b)^n = a^n \cdot b^n$
raise each factor separately
Zero and negative exponents
$a^0 = 1, \quad a^{-n} = \frac{1}{a^n} \quad (a \neq 0)$
a negative exponent means a reciprocal

A power is shorthand for multiplication of the same factor by itself:

a^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ factors}}

The number $a$ is the base and $n$ is the exponent — it says how many times the base appears as a factor:

2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32

The second power is called a square ( $5^2$ reads "five squared") and the third one a cube ( $5^3$ is "five cubed").

The laws of exponents

Every law follows straight from the definition — write the power out as a product and count the factors.

Multiplying powers with the same base adds the exponents:

a^m \cdot a^n = a^{m+n}

because $2^3 \cdot 2^2 = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2) = 2^5$ .

Dividing subtracts them:

\frac{a^m}{a^n} = a^{m-n}

And raising a power to a power multiplies them:

(a^m)^n = a^{m \cdot n}

Write as a single power: 3⁴ · 3³ ÷ 3²

Raising to a power distributes over multiplication and division, but not over addition:

(a \cdot b)^n = a^n \cdot b^n

By contrast $(a + b)^n \neq a^n + b^n$ — check it on $(1 + 2)^2 = 9$ while $1^2 + 2^2 = 5$ .

Zero and negative exponents

What does $a^0$ mean? "Multiply the base by itself zero times" says nothing, so the answer comes from the division law:

a^0 = a^{n-n} = \frac{a^n}{a^n} = 1 \quad (a \neq 0)

So $a^0 = 1$ is not an arbitrary convention — it is the only value that keeps the division law true. Negative exponents follow the same track:

a^{-n} = \frac{1}{a^n} \quad (a \neq 0)

A negative exponent means a reciprocal, not a negative result: $2^{-3} = \frac{1}{8}$ is a positive number.

What is 5⁻² ?

Scientific notation

Powers of ten keep very large and very small numbers short. In scientific notation a number is written as $a \cdot 10^n$ with $1 \le a < 10$ :

300\,000\,000 = 3 \cdot 10^8 \qquad 0.00042 = 4.2 \cdot 10^{-4}

The exponent says how far the decimal point moves: to the right when positive, to the left when negative.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

2² =

Common mistakes

Multiplying the base by the exponent — $2^5$ is $32$ , not $10$ ; the exponent counts factors, it does not multiply.
Adding exponents when powers are added — the laws cover multiplication and division; $2^3 + 2^4$ has to be computed as $8 + 16 = 24$ .
Dropping the bracket on a negative base — $(-2)^4 = 16$ , but $-2^4 = -16$ .
Reading $2^{-3}$ as a negative number — a negative exponent gives a reciprocal, that is $\frac{1}{8}$ .

Formula card

Topic: Powers

Definition of a power
$a^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ factors}}$
a is the base, n is the exponent
Multiplying powers with the same base
$a^m \cdot a^n = a^{m+n}$
add the exponents
Dividing powers with the same base
$\frac{a^m}{a^n} = a^{m-n}$
subtract the exponents
A power of a power
$(a^m)^n = a^{m \cdot n}$
multiply the exponents
A power of a product
$(a \cdot b)^n = a^n \cdot b^n$
raise each factor separately
Zero and negative exponents
$a^0 = 1, \quad a^{-n} = \frac{1}{a^n} \quad (a \neq 0)$
a negative exponent means a reciprocal

Frequently asked questions

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11 min