The Laws of Exponents: Product, Quotient and Power Rules

Q: When do you add exponents and when do you multiply them?

Add when you multiply two powers of the same base: a^m × a^n = a^m+n. Multiply when you raise a power to a power: (a^m)^n = a^mn.

Q: Can you add exponents when the bases are different?

No. 2^3 × 5^2 stays as it is (or is worked out as 8 × 25). The laws only combine powers of the same base.

Every exponent law follows from counting factors. This article builds the product, quotient and power laws from that one idea, and shows the mistakes that come from using them on different bases or on sums.

3 min read Supports 4 lessons

You do not need to memorise the exponent laws. Each one is what you get by writing the powers out in full and counting how many factors there are.

Multiplying powers: add the exponents

$2^3 \times 2^4$ is three $2$ s multiplied by four more $2$ s. That is seven $2$ s in a row.

$2^3 \times 2^4$
Three 2s, then four 2s.

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Product law

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

Dividing powers: subtract the exponents

In $\frac{2^6}{2^2}$ , two of the $2$ s on top cancel with the two on the bottom, leaving four.

\frac{2^6}{2^2} = \frac{\cancel{2} \cdot \cancel{2} \cdot 2 \cdot 2 \cdot 2 \cdot 2}{\cancel{2} \cdot \cancel{2}} = 2^4

Quotient law

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

A power of a power: multiply

$(5^2)^3$ means $5^2$ three times over: $5^2 \times 5^2 \times 5^2$ . Each holds two $5$ s, so there are $2 \times 3 = 6$ of them.

Power law

(a^m)^n = a^{mn} \qquad (ab)^n = a^n b^n \qquad \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Putting them together

Worked example

Simplify $\dfrac{x^7 \cdot x^2}{x^4}$ .

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Product law on top.
$\frac{x^{9}}{x^4}$

Worked example

Simplify $(3x^2y)^3$ .

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Every factor inside is cubed.
$3^3 \cdot (x^2)^3 \cdot y^3$

Where the laws do not apply

Watch out

The laws need the same base. $2^3 \cdot 3^2$ cannot be combined into a single power, since the factors are different.

Watch out

They work for products, not sums. $(a + b)^2 \neq a^2 + b^2$ . Try $a = b = 1$ : the left is $4$ and the right is $2$ .

Watch out

$x^2 + x^3$ is not $x^5$ . Adding powers does not add exponents; only multiplying them does.

What the laws lead to

If the laws are to keep working, they force some surprising definitions. The quotient law with equal exponents gives $a^0 = 1$ . With a bigger exponent underneath it gives negative powers, and the power law makes $a^{1/2}$ a square root. Later, the log laws turn out to be these same laws seen from the other side.

Tip

Stuck? Write the powers out in full with small numbers. The pattern will tell you which law applies.

Common questions

When do you add exponents and when do you multiply them?: Add when you multiply two powers of the same base: $a^m \cdot a^n = a^{m+n}$ . Multiply when you raise a power to a power: $(a^m)^n = a^{mn}$ .
Can you add exponents when the bases are different?: No. $2^3 \times 5^2$ stays as it is (or is worked out as $8 \times 25$ ). The laws only combine powers of the same base.

Practise exponent laws

The The Exponent Laws lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

When do you add exponents and when do you multiply them?

Add when you multiply two powers of the same base: $a^m \cdot a^n = a^{m+n}$ . Multiply when you raise a power to a power: $(a^m)^n = a^{mn}$ .

Can you add exponents when the bases are different?

No. $2^3 \times 5^2$ stays as it is (or is worked out as $8 \times 25$ ). The laws only combine powers of the same base.