Power of a power: what it means and how to use it
Raise an existing power by multiplying exponents.
Power of a power is one of 3 exponents formulas in the algebra section of this library, and it is used at high school level.
Why power of a power works
The outer exponent says how many copies of the inner power to write down, and each copy supplies m factors of a. Counting the total is like counting a rectangle with n rows of m dots, which is multiplication rather than addition.
What each symbol means
$a$ is the base and $m,n$ are exponents.
Power of a power: when it holds
A power of a product distributes, but a power of a sum generally does not.
When it stops applying
The rule covers a power of a power and a power of a product, but never a power of a sum. Squaring 3 + 4 gives 49, while squaring the pieces separately and adding gives 9 plus 16, which is 25.
Power of a power: a worked example
$(x^3)^4=x^{12}$.
The mistake to avoid
What people do: Adding the exponents and writing x cubed raised to the fourth as x to the seventh.
Why it goes wrong: Test it with a number: 2 cubed squared is 8 squared, which is 64, and that is 2 to the sixth, not 2 to the fifth which is 32. Adding undercounts every factor the repetition creates.
Do this instead: Ask how many groups and how many in each group. Four groups of three factors is twelve factors, so the answer is x to the twelfth.
Power of a power: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a$ is the base and $m,n$ are exponents.
- Check the conditions before substituting. A power of a product distributes, but a power of a sum generally does not.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Exponents
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where power of a power comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms lesson — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about power of a power
Is a bracketed power the same as a stacked one?
No, and the brackets decide it. Try base 2 with exponents 3 and 2: the bracketed version 2 cubed then squared is 64, while the stacked version 2 raised to 3 squared is 2 to the ninth, which is 512.
Does the rule survive negative exponents?
Yes. Raising x to the -2 to the third power multiplies -2 by 3, giving x to the -6, which is the same as 1 over x to the sixth.
How is (2x) cubed different?
Everything inside the bracket gets cubed, including the 2. So (2x) cubed is 8x cubed, whereas 2x cubed only cubes the x and leaves the 2 alone.
Can the outer exponent be a fraction?
Yes, but watch the sign of the base. Raising x to the sixth to the power one half gives the absolute value of x cubed, since the result of an even root can never be negative.