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^½)² = b the reason √b deserves the name b^½ in the first place.

The commonest error is confusing this law with the product law: b^m · 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) = 2⁹ = 512, not 2⁶. 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.