Power of a Power

Q: What if the problem mixes both rules, like $(2^{4})^{3} \cdot 2^{2}$?

Handle the power of a power first: $(2^{4})^{3} = 2^{12}$. Then the product rule adds the remaining exponents: $2^{12} \cdot 2^{2} = 2^{14}$. Inside-out order keeps it clean.

A power of a power is an expression like $(5^{3})^{4}$ — a power sitting inside parentheses with another exponent on the outside. The rule is short: multiply the exponents and keep the base. So $(5^{3})^{4} = 5^{3 \cdot 4} = 5^{12}$ .

The whole trick to this lesson is knowing when to multiply exponents and when to add them. Parentheses with an exponent outside means multiply. Two powers of the same base multiplied side by side means add. Mixing those two up is the single most common error on this topic, so we will hit the difference head-on.

Why you multiply the exponents

Take $(2^{3})^{4}$ . The outside exponent $4$ says to write $2^{3}$ as a factor four times: $2^{3} \cdot 2^{3} \cdot 2^{3} \cdot 2^{3}$ . Each $2^{3}$ carries three factors of $2$ , and there are four groups of them — that is $3 \cdot 4 = 12$ factors of $2$ in total.

So $(2^{3})^{4} = 2^{12}$ , and in general $(2^{a})^{b} = 2^{a \cdot b}$ . The rule is not something to memorize on faith — it is just counting factors in groups.

Multiply here, add there

Compare these two expressions carefully: $2^{3} \cdot 2^{4} = 2^{7}$ , but $(2^{3})^{4} = 2^{12}$ . The first is a product of powers — same base multiplied side by side — so the exponents add. The second is a power of a power — parentheses with an exponent outside — so the exponents multiply.

When you see both in one problem, simplify the power of a power first, then apply the product rule to what is left.

The base does not change

In $(7^{2})^{5}$ , only the exponents interact: the answer is $7^{10}$ , with the base still $7$ . Wrong answers on this topic almost always touch the base — turning $(7^{2})^{5}$ into $14^{10}$ or $49^{5}$ -style rewrites. Leave the base alone, multiply the exponents, and you are done.

Worked examples

Example 1: apply the rule

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

Power of a power: multiply the exponents

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

Multiply

3 \cdot 2 = 6

Write the result

(2^{3})^{2} = 2^{6}

Answer: $2^{6}$

Example 2: a base of 10

Write $(10^{4})^{3}$ as a single power.

Power of a power: multiply the exponents

(10^{4})^{3} = 10^{4 \cdot 3}

Multiply

4 \cdot 3 = 12

Write the result

(10^{4})^{3} = 10^{12}

Answer: $10^{12}$

Example 3: simplify, then evaluate

Evaluate $(3^{2})^{2}$ .

Power of a power: multiply the exponents

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

Write out the factors

3^{4} = 3 \cdot 3 \cdot 3 \cdot 3

Multiply

3^{4} = 81

Answer: $81$

Try one yourself

Common questions

Do I add or multiply the exponents?

Multiply when a power is raised to another power: $(2^{3})^{4} = 2^{12}$ . Add when powers of the same base are multiplied side by side: $2^{3} \cdot 2^{4} = 2^{7}$ . The parentheses with an exponent outside are your signal to multiply.

Does the base change?

No. In $(7^{2})^{5} = 7^{10}$ the base stays $7$ the entire time. Only the exponents interact. If your answer has a different base than the problem started with, something went wrong.

What if the problem mixes both rules, like

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

Handle the power of a power first: $(2^{4})^{3} = 2^{12}$ . Then the product rule adds the remaining exponents: $2^{12} \cdot 2^{2} = 2^{14}$ . Inside-out order keeps it clean.

Does the rule work with negative exponents?

Yes — multiply exactly as before, watching signs. For example, $(2^{-3})^{4} = 2^{-12}$ , because $-3 \cdot 4 = -12$ . The rule itself never changes.

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Power of a Power

Why you multiply the exponents

Multiply here, add there

The base does not change

Worked examples

Example 1: apply the rule

Example 2: a base of 10

Example 3: simplify, then evaluate

Try one yourself

Common questions

Keep learning

Exponent Notation & Meaning

Zero & Negative Exponents

Product & Quotient Rules

Operations in Scientific Notation