Power Rule

Q: Does the outside exponent apply to the number in front?

Yes. In $(4x^{3})^{2}$, both the $4$ and the $x^{3}$ get squared: $16x^{6}$. Skipping the coefficient is the single most common error on these problems.

When you raise a power to another power, you multiply the exponents: $(x^{4})^{3} = x^{12}$ . That's the power rule, sometimes called power of a power.

Why multiply? $(x^{4})^{3}$ means three copies of $x^{4}$ multiplied together: $x^{4} \cdot x^{4} \cdot x^{4}$ . By the multiplication rule that's $x^{4+4+4} = x^{12}$ — and adding $4$ three times is exactly $4 \cdot 3$ . The power rule is repeated use of the multiplication rule, compressed into one step.

Power of a power: multiply

In symbols, $(x^{a})^{b} = x^{a \cdot b}$ . Keep the base, multiply the exponents.

Keep this straight against the multiplication rule: multiplying two powers adds exponents ( $x^{4} \cdot x^{3} = x^{7}$ ), while raising a power to a power multiplies them ( $(x^{4})^{3} = x^{12}$ ). If you're ever unsure which is which, test with small numbers like $2^{2}$ and $2^{3}$ .

Power of a product: the exponent reaches every factor

When a whole product is raised to a power, the exponent applies to every factor inside the parentheses — including the coefficient. In symbols, $(xy)^{a} = x^{a}y^{a}$ .

So $(2x^{4}y)^{3}$ becomes $2^{3} \cdot (x^{4})^{3} \cdot y^{3} = 8x^{12}y^{3}$ . The most common mistake here is leaving the coefficient alone and writing $2x^{12}y^{3}$ — the $2$ gets cubed too.

The zero power

Any nonzero base raised to the $0$ power equals $1$ : $x^{0} = 1$ and $10^{0} = 1$ . You can see why from the division rule: $\dfrac{x^{3}}{x^{3}} = x^{3-3} = x^{0}$ , and anything divided by itself is $1$ .

Worked examples

Example 1: power of a power

Simplify $(x^{4})^{3}$ .

Power to a power, so multiply the exponents

x^{4 \cdot 3}

Multiply

x^{12}

Answer: $x^{12}$

Example 2: the exponent hits the coefficient too

Simplify $(2x^{4}y)^{3}$ .

Cube every factor inside the parentheses

2^{3} \cdot (x^{4})^{3} \cdot y^{3}

Cube the coefficient

2^{3} = 8

Multiply the exponents on

x

4 \cdot 3 = 12

Put it together

8x^{12}y^{3}

Answer: $8x^{12}y^{3}$

Example 3: combine with the multiplication rule

Simplify $(3y^{2})^{2} \cdot (y^{3})^{4}$ .

Apply the power rule to each group

9y^{4} \cdot y^{12}

Now multiply — same base, add the exponents

9y^{4+12}

Add

9y^{16}

Answer: $9y^{16}$

Try one yourself

Common questions

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

Multiplying two powers adds the exponents: $x^{4} \cdot x^{3} = x^{7}$ . Raising a power to a power multiplies them: $(x^{4})^{3} = x^{12}$ . Look for the parentheses with an exponent outside — that's the multiply case.

Does the outside exponent apply to the number in front?

Yes. In $(4x^{3})^{2}$ , both the $4$ and the $x^{3}$ get squared: $16x^{6}$ . Skipping the coefficient is the single most common error on these problems.

Why does anything to the zero power equal 1?

Divide a power by itself: $\dfrac{x^{3}}{x^{3}}$ is $1$ because the top and bottom match, and it's also $x^{3-3} = x^{0}$ by the division rule. For the rules to stay consistent, $x^{0}$ must be $1$ (for any nonzero $x$ ).

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Power Rule

Power of a power: multiply

Power of a product: the exponent reaches every factor

The zero power

Worked examples

Example 1: power of a power

Example 2: the exponent hits the coefficient too

Example 3: combine with the multiplication rule

Try one yourself

Common questions

Keep learning

Multiplication Rule

Division Rule

Negative Exponent Rule

Power of a Fraction and Rational Exponents