Power of a Power Calculator

The Power of a Power Property

In algebra, the power of a power rule states that when an exponential term is raised to another power, you multiply the exponents while keeping the base identical:

$$(b^m)^n = b^{m \cdot n}$$

Where $b$ is the base, $m$ is the inner exponent, and $n$ is the outer exponent. For example, $(2^3)^4$ means $(2^3)$ multiplied by itself 4 times: $(2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) \times (2 \times 2 \times 2) = 2^{12} = 4,096$. Multiplying the exponents $3 \times 4 = 12$ gives the answer immediately.

Extending to Multiple Nested Exponents

The power of a power property holds for any number of nested exponents:

$$\left(\left(b^m\right)^n\right)^p = b^{m \cdot n \cdot p}$$

For instance, $((2^2)^3)^2 = 2^{2 \times 3 \times 2} = 2^{12} = 4,096$.

Negative and Fractional Exponents

The rule applies seamlessly when exponents are negative or fractional:

• Negative exponents: $(3^{-2})^3 = 3^{(-2) \times 3} = 3^{-6} = \frac{1}{3^6} = \frac{1}{729}$.
• Fractional exponents: $(4^{1/2})^3 = 4^{(1/2) \times 3} = 4^{3/2} = (\sqrt{4})^3 = 2^3 = 8$.

Power of a Power vs Power Towers (Order of Operations)

One of the most common pitfalls in algebra is confusing $(b^m)^n$ with $b^{m^n}$.

Parentheses force evaluation from inside out:

$$(2^3)^4 = 2^{3 \times 4} = 2^{12} = 4,096$$

Without parentheses, standard mathematical convention dictates that chained exponents evaluate from top to bottom (right to left):

$$2^{3^4} = 2^{(3^4)} = 2^{81} \approx 2.41785 \times 10^{24}$$

As you can see, $2^{12}$ and $2^{81}$ differ by 21 orders of magnitude! Always check whether grouping parentheses are present.

For other exponent laws, try the Multiplying Exponents Calculator, the Dividing Exponents Calculator, and the general Exponent Calculator.