A power of a power is what happens when you take a number that’s already raised to an exponent and raise the whole thing to another exponent. The rule is straightforward: you multiply the two exponents together. So (xa)b equals xa×b. That single formula covers every case you’ll run into.
How the Rule Works
Say you have (32)4. The inner exponent tells you 3 is multiplied by itself twice (3 × 3 = 9). The outer exponent tells you to take that result and multiply it by itself four times (9 × 9 × 9 × 9 = 6,561). Using the power of a power rule, you can skip straight to the answer: multiply the exponents 2 × 4 = 8, giving you 38, which is also 6,561.
The reason this works becomes obvious if you write it out the long way. (32)4 means four copies of 32 multiplied together: 32 × 32 × 32 × 32. Each of those contains two 3s being multiplied, so you end up with 3 multiplied by itself 2 + 2 + 2 + 2 = 8 times. Adding the same number four times is the same as multiplying it by four. That’s why the shortcut is to multiply the exponents.
Examples With Variables
The rule works the same way whether you’re dealing with plain numbers or variables:
- (x3)5 = x3×5 = x15
- (y4)2 = y4×2 = y8
- (25)3 = 25×3 = 215 = 32,768
If there’s a coefficient (a number in front of the variable), the outer exponent applies to it as well. For example, (2x3)4 means you raise both the 2 and the x3 to the fourth power: 24 × x3×4 = 16x12.
Negative and Fractional Exponents
The multiply-the-exponents rule applies to every kind of exponent, not just whole positive numbers.
With negative exponents: (x-2)3 = x-2×3 = x-6, which is the same as 1/x6. The multiplication follows normal rules for multiplying negative numbers, so (x-3)-2 = x-3×-2 = x6. A negative times a negative gives a positive exponent.
With fractions: (x1/2)4 = x(1/2)×4 = x2. This is useful because x1/2 is the same as the square root of x, so this tells you that the square root of x, raised to the fourth power, equals x2. Similarly, (x2/3)6 = x(2/3)×6 = x4.
The Mistake Almost Everyone Makes
The most common error is mixing up the power of a power rule with the product rule for exponents. Here’s the difference:
- Multiplying same-base terms: x2 × x3 = x2+3 = x5 (you ADD the exponents)
- Power of a power: (x2)3 = x2×3 = x6 (you MULTIPLY the exponents)
The parentheses are what tell you which rule to use. When two exponential terms are sitting next to each other being multiplied, you add exponents. When one exponent is wrapped in parentheses with another exponent outside, you multiply. Getting these two confused is the single most frequent mistake in algebra involving exponents.
Why Parentheses Matter
Without parentheses, the expression means something completely different. Compare these two:
- (23)2 = 26 = 64 (power of a power, multiply exponents)
- 232 = 29 = 512 (exponent tower, evaluate from top down)
In the second expression, with no parentheses, you evaluate the exponents from right to left: first compute 32 = 9, then compute 29 = 512. Standard order of operations says you handle innermost groupings first, then work outward. This is why notation matters. The parentheses in (23)2 explicitly tell you to treat 23 as a single unit and then square it.
Combining Multiple Exponent Rules
In practice, you’ll often need the power of a power rule alongside other exponent rules in the same problem. Take an expression like (x2y3)4. The outer exponent distributes to every factor inside the parentheses: x2×4 × y3×4 = x8y12.
A slightly more complex example: simplify (x3)2 × x4. First, apply the power of a power rule to get x6 × x4. Then use the product rule (add exponents when multiplying same-base terms) to get x10. Breaking the problem into small steps and using one rule at a time keeps errors to a minimum.
For division, the same layered approach works. To simplify (x5)3 / x7, start with the power of a power: x15 / x7. Then subtract exponents (the quotient rule): x15-7 = x8.