Product & Quotient Rules

Q: Is $2^{-3}$ a negative number?

No. $2^{-3} = \dfrac{1}{2^{3}} = \dfrac{1}{8}$, a positive fraction. The negative sign in the exponent means the reciprocal — flip the power into the bottom of a fraction — not that the value is below zero.

An exponent counts how many times a base is used as a factor: $3^{4}$ means $3 \cdot 3 \cdot 3 \cdot 3$ . The exponent rules are shortcuts that fall straight out of that definition. Instead of writing out long strings of factors, you add, subtract, or multiply the exponents.

There are really only three moves to learn — multiply powers, divide powers, raise a power to a power — plus two special cases, the zero exponent and negative exponents. Every rule below only works when the bases match, so check that first every single time.

Product rule and quotient rule

Product rule: when you multiply powers with the same base, keep the base and add the exponents. $x^{a} \cdot x^{b} = x^{a+b}$ . Why? $3^{5} \cdot 3^{2}$ is five factors of $3$ next to two more factors of $3$ — that is seven factors of $3$ in total, so $3^{7}$ .

Quotient rule: when you divide powers with the same base, keep the base and subtract the exponents. $\dfrac{x^{a}}{x^{b}} = x^{a-b}$ . The factors on the bottom cancel matching factors on top, and whatever is left over is the answer.

The classic trap: the base does not change. $3^{5} \cdot 3^{2}$ is $3^{7}$ , not $9^{7}$ . You are counting factors of $3$ , not multiplying the bases together.

Power rule

Power rule: when a power is raised to another power, keep the base and multiply the exponents. $(x^{a})^{b} = x^{ab}$ . Something like $(2^{3})^{4}$ means four copies of $2^{3}$ multiplied together — that is $3 + 3 + 3 + 3 = 12$ factors of $2$ , which is exactly $3 \cdot 4$ .

A quick way to keep the product rule and power rule straight: multiplying powers adds exponents; a power of a power multiplies them. If you are ever unsure, write out a small case with actual factors — the rule will reappear on its own.

Zero and negative exponents

Zero exponent: any nonzero base to the zero power equals $1$ . That is, $x^{0} = 1$ . It has to be — the quotient rule says $\dfrac{5^{3}}{5^{3}} = 5^{0}$ , and any number divided by itself is $1$ .

Negative exponent: a negative exponent means a reciprocal, not a negative number. $x^{-n} = \dfrac{1}{x^{n}}$ . So $2^{-3} = \dfrac{1}{2^{3}} = \dfrac{1}{8}$ , which is a small positive fraction. If a subtraction in the quotient rule leaves you with a negative exponent, just move the power to the bottom of a fraction.

Worked examples

Example 1: product rule

Write $2^{4} \cdot 2^{3}$ as a single power, then evaluate it.

Same base, so add the exponents

2^{4} \cdot 2^{3} = 2^{4+3} = 2^{7}

Evaluate

2^{7} = 128

Check with the definition:

16 \cdot 8 = 128

✓

Answer: $2^{7} = 128$

Example 2: quotient rule

Simplify $\dfrac{5^{7}}{5^{4}}$ .

Same base, so subtract the exponents

\dfrac{5^{7}}{5^{4}} = 5^{7-4} = 5^{3}

Evaluate

5^{3} = 125

Answer: $5^{3} = 125$

Example 3: power rule

Write $(3^{2})^{4}$ as a single power.

A power of a power, so multiply the exponents

(3^{2})^{4} = 3^{2 \cdot 4} = 3^{8}

Check:

(3^{2})^{4} = 9^{4} = 6561

and

3^{8} = 6561

✓

Answer: $3^{8}$

Example 4: a negative exponent appears

Simplify $2^{3} \cdot 2^{-5}$ and write the answer as a fraction.

Same base, so add the exponents

2^{3} \cdot 2^{-5} = 2^{3+(-5)} = 2^{-2}

A negative exponent means a reciprocal

2^{-2} = \dfrac{1}{2^{2}}

Evaluate

\dfrac{1}{2^{2}} = \dfrac{1}{4}

Check:

8 \cdot \dfrac{1}{32} = \dfrac{8}{32} = \dfrac{1}{4}

✓

Answer: $2^{-2} = \dfrac{1}{4}$

Try one yourself

Common questions

Do the exponent rules work when the bases are different?

No. $2^{3} \cdot 5^{2}$ cannot be combined into a single power, because the factors are different numbers. Just evaluate each power and multiply: $8 \cdot 25 = 200$ . The rules only combine powers of the same base.

2^{-3}

a negative number?

No. $2^{-3} = \dfrac{1}{2^{3}} = \dfrac{1}{8}$ , a positive fraction. The negative sign in the exponent means the reciprocal — flip the power into the bottom of a fraction — not that the value is below zero.

Why does anything to the zero power equal

1

Follow the quotient rule: $\dfrac{4^{2}}{4^{2}} = 4^{2-2} = 4^{0}$ . But $\dfrac{16}{16} = 1$ , so $4^{0}$ must be $1$ . The same argument works for any nonzero base.

Ready for more practice? Print the Properties of Exponents worksheet with an answer key.

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Product & Quotient Rules

Product rule and quotient rule

Power rule

Zero and negative exponents

Worked examples

Example 1: product rule

Example 2: quotient rule

Example 3: power rule

Example 4: a negative exponent appears

Try one yourself

Common questions

Keep learning

Writing Numbers in Scientific Notation

Perfect Squares & Square Roots

Simplifying Radical Expressions