Power Rule for Exponents

The power rule for exponents tells you what happens when a power is itself raised to another power. In symbols, for any nonzero base \(a\) and exponents \(x\) and \(y\):

\((a^x)^y = a^{xy}\)

In words, when you raise a power to a power, you keep the same base and multiply the exponents. This is one of the core laws of exponents that shows up constantly once you start simplifying algebraic expressions, and it is worth knowing well before moving into more mixed problems that combine several exponent rules at once.

The rule is not just a shortcut to memorize, it comes directly from what an exponent means. Consider \((3^2)^4\). The outer exponent of 4 means you multiply \(3^2\) by itself four times:

\((3^2)^4 = 3^2 \times 3^2 \times 3^2 \times 3^2\)

Each \(3^2\) contributes two factors of 3, and there are four groups of them, so altogether there are \(2 \times 4 = 8\) factors of 3:

\(3^2 \times 3^2 \times 3^2 \times 3^2 = 3^{8}\)

That matches the power rule directly: \((3^2)^4 = 3^{2 \times 4} = 3^8\). The same reasoning works for any base and any whole-number exponents, which is why the rule holds in general.

Example 1: Simplify \((5^3)^2\).

Multiply the exponents, keeping the base the same: \((5^3)^2 = 5^{3 \times 2} = 5^6\).

Example 2: Simplify \((x^4)^5\).

The base is \(x\), so \((x^4)^5 = x^{4 \times 5} = x^{20}\).

Example 3: Simplify \((2^{-3})^2\).

The power rule still applies with a negative exponent: \((2^{-3})^2 = 2^{-3 \times 2} = 2^{-6}\). If you need a refresher on what a negative exponent means once you land on an answer like this, see how to simplify negative exponents.

Example 4: Simplify \((y^2)^0\).

Multiplying the exponents gives \(y^{2 \times 0} = y^0 = 1\), since any nonzero base raised to the zero power equals 1.

The power rule also means that \((2^x)^2\) and \(4^x\) are actually the same function, since \((2^x)^2 = 2^{2x} = (2^2)^x = 4^x\). The graph below plots \(y = 4^x\), which is exactly what you get after applying the power rule to \((2^x)^2\).

The power rule is one piece of a larger toolkit. When you multiply two identical bases, you add exponents instead of multiplying them, which is a different rule entirely. When a power applies to a product inside parentheses, such as \((2x)^3\), you distribute the exponent to each factor using the power of a product rule. Many algebra problems mix these ideas together, so it helps to be comfortable applying more than one rule in the same expression once you have the power rule down.

A frequent error is adding the exponents instead of multiplying them, mixing up the power rule with the product rule. Remember: \((a^x)^y\) means multiply, while \(a^x \times a^y\) means add. Another common mistake is applying the outer exponent only to part of an expression. In \((3x)^2\), the exponent applies to both the 3 and the \(x\), not just the \(x\), so \((3x)^2 = 9x^2\), not \(3x^2\). Always check exactly what is enclosed by the parentheses before multiplying exponents.

Try simplifying these using the power rule before checking your work: \((4^2)^3\), \((x^5)^2\), \((a^{-2})^3\), and \((y^0)^7\). In each case, keep the base unchanged and multiply the exponents together to reach the simplified form.

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