Explaining maths

$\def\N{\mathbb N}$ $\def\F{\mathbb F}$ $\def\R{\mathbb R}$ $\def\C{\mathbb C}$ $\def\Q{\mathbb Q}$ $\def\Z{\mathbb Z}$ $\def\jq{\;\,}$ $\def\rd{\textrm{d}}$ $\def\Rbar{{\overline{\mathbb{R}}}}$ $\def\Pset{{\mathcal{P}}}$ $\def\Jds{\displaystyle}$ $\def\Jand{\quad\textrm{and}\quad}$ $\def\Jdisplay{\qquad\qquad\qquad\displaystyle}$ $\def\ve{\varepsilon}$ $\def\jnorm{\|\cdot\|}$ $\def\op{\mathrm{op}}$ $\def\RHS{\mathrm{RHS}}$ $\def\LHS{\mathrm{LHS}}$ $\def\lin{\mathop{\mathrm{lin}}}$ $\def\i{\mathrm{i}}$ $\def\e{\mathrm{e}}$ $\def\Re{\mathop{\mathrm{Re}}}$ $\def\Im{\mathop{\mathrm{Im}}}$ $\def\epsilon{\varepsilon}$ $\def\va{\boldsymbol{a}}$ $\def\vb{\boldsymbol{b}}$ $\def\vx{\boldsymbol{x}}$ $\def\vy{\boldsymbol{y}}$ $\def\vv{\boldsymbol{v}}$ $\def\Ve{\boldsymbol{e}}$ Explaining maths

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Problems when multiplying powers

- February 10, 2026

In the past I used to confuse students by asking them what the correct value of $((-1)^{2/3})^{3/2}$ is. This is on the assumption that you follow the convention that $x^{2/3}$ is equal to the square of the cube root of $x$ for all real numbers $x$. But perhaps instead I should argue that $1$ must be equal to $\pm \mathrm{i}$, on the grounds that $((-1)^2)^{1/4}$ should be equal to $(-1)^{1/2}$?