Power of a Power

A power raised to a power multiplies the exponents, because $(b^m)^n$ means n copies of $b^m$ and each contributes m factors of b. Worked example: (2³)⁴ = 2¹² = 4096 — twelve twos, however you group them. The rule holds for negative and fractional exponents too, which is what makes $(b^{1/2})^2 = b$ the reason √b deserves the name $b^{1/2}$ in the first place.

The commonest error is confusing this law with the product law: $b^m \cdot b^n$ adds the exponents, while $(b^m)^n$ multiplies them, so 2³ · 2⁴ = 2⁷ = 128 but (2³)⁴ = 2¹² = 4096. A second trap is that a stacked power without brackets is read top-down, so $2^{(3^2)} = 2^{9} = 512$, not $2^{6}$. This calculator keeps the base positive so that fractional exponents remain real: $(-8)^{1/3}$ is −2 if you insist on real cube roots, but $(-8)^{2/6}$ would then have to be +2, and the exponent laws quietly break. Chemists and computer scientists lean on the rule constantly: (10³)³ = 10⁹ converts cubic metres to cubic millimetres in one step, and doubling the exponent doubles the number of digits in binary.