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.

How do you raise a fraction to a power?

Raise the numerator and the denominator separately: (a/b)ⁿ = aⁿ/bⁿ, for b ≠ 0. For example (2/3)³ = 2³/3³ = 8/27. It is the same law as the power of a product, since dividing is multiplying by the reciprocal.

Does a larger base always give a larger power?

With the same exponent and non-negative bases, yes: 0 ≤ a 0.5⁵). When the bases AND the exponents differ, no rule is enough — you have to compare the values, as in the pair 3⁵ = 243 and 5³ = 125.

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, powers of a product and of a quotient, zero and negative exponents, the monotonicity of exponentiation, 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.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Drives and memory sticks
A "64 GB" stick shows up in the file explorer as 59.6 GB: the maker counts 64 · 10⁹ bytes, the system divides by 2³⁰ = 1,073,741,824. Nobody stole the missing 4.4 GB — two different powers did.
Convert it here: Data (IEC)
Saving money
Compound interest is a power, not a product: 10,000 at 4% a year grows to 10,000 · 1.04¹⁰ = 14,802 over a decade, not the 14,000 that "4% ten times over" suggests.
Food safety
Bacteria in a warm dish divide every 20 minutes, so after n hours there are 2³ⁿ times as many. Two hours on the counter is 2⁶ = 64 times the cells, six hours is 2¹⁸ = 262,144 times — four thousand times worse than the two-hour figure. That is why the rules set a two-hour cooling window instead of saying "do not leave it out overnight": inside an exponent every extra hour costs eight times as much.

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
A power of a quotient
$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)$
raise the numerator and the denominator 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
Monotonicity in the base
$0 \le a < b \implies a^n < b^n \quad (n \ge 1)$
with the same exponent, the larger base gives the larger power
Monotonicity in the exponent
$m < n \implies \begin{cases} a^m < a^n & \text{for } a > 1 \\ a^m > a^n & \text{for } 0 < a < 1 \end{cases}$
a base above 1 makes the power grow, a base between 0 and 1 makes it shrink

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 \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)

Both are one law: dividing is multiplying by the reciprocal, so a quotient is raised to a power by raising the numerator and the denominator separately. Writing it out shows why:

\left(\frac{2}{3}\right)^3 = \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} = \frac{2 \cdot 2 \cdot 2}{3 \cdot 3 \cdot 3} = \frac{2^3}{3^3} = \frac{8}{27}

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

Compute (3/5)² and (1/2)⁴.

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.

For a fraction the two laws meet in one handy consequence: $\left(\frac{a}{b}\right)^{-n} = \frac{a^{-n}}{b^{-n}} = \frac{b^n}{a^n}$ , that is, flip the fraction and raise it. Hence $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$ .

What is 5⁻² ?

The monotonicity of exponentiation

Powers can be compared without working out their values — provided only one thing about them differs.

When the exponent is the same, the base decides. Among non-negative bases, the larger base gives the larger power:

0 \le a < b \implies a^n < b^n \quad (n \ge 1)

That settles $3^4 < 4^4$ and $\left(\frac{1}{3}\right)^5 < \left(\frac{1}{2}\right)^5$ straight away, with nothing computed.

When the base is the same, the exponent decides — but the direction depends on whether that base is above or below one:

m < n \implies \begin{cases} a^m < a^n & \text{for } a > 1 \\ a^m > a^n & \text{for } 0 < a < 1 \end{cases}

The reason takes one sentence: every further power multiplies by the base, and multiplying by a number above $1$ makes things bigger while multiplying by one between $0$ and $1$ makes them smaller. Hence $2^3 < 2^5$ ( $8 < 32$ ), but $0.5^3 > 0.5^5$ ( $0.125 > 0.03125$ ). A base of $1$ is the boundary where nothing changes at all: $1^n = 1$ for every $n$ .

When the base and the exponent both differ, neither rule is enough — you have to bring the powers to a common form or compute the values. The pair $3^5$ and $5^3$ shows that intuition fails here: the larger base loses, since $3^5 = 243$ while $5^3 = 125$ . There is more force in the exponent than in the base.

Which is larger: 2¹⁰ or 10²? And 0.9¹⁰ or 0.9³?

Monotonicity is also why exponentiation can be undone: since a power with a fixed base above $1$ grows with its exponent, every value corresponds to exactly one exponent — and that exponent is what a logarithm looks for.

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.

An exponent need not be a whole number: $a^{m/n}$ makes sense and means exactly $\sqrt[n]{a^m}$ — that is the subject of its own topic, rational exponents.

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

4² =

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 $\frac{1}{8}$ .
Raising only the numerator — $\left(\frac{2}{3}\right)^3$ is $\frac{8}{27}$ , not $\frac{8}{3}$ ; the denominator is raised just like the numerator.
Assuming a higher exponent always gives a bigger number — with a base between $0$ and $1$ it is the other way round: $0.5^5 < 0.5^3$ .
Comparing powers "by the base" when the exponents differ too — $5^3 = 125$ is smaller than $3^5 = 243$ .

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
A power of a quotient
$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)$
raise the numerator and the denominator 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
Monotonicity in the base
$0 \le a < b \implies a^n < b^n \quad (n \ge 1)$
with the same exponent, the larger base gives the larger power
Monotonicity in the exponent
$m < n \implies \begin{cases} a^m < a^n & \text{for } a > 1 \\ a^m > a^n & \text{for } 0 < a < 1 \end{cases}$
a base above 1 makes the power grow, a base between 0 and 1 makes it shrink

Frequently asked questions

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