**Definition** We have shown that [[Existence & uniqueness nth root of positive reals]]. If $\alpha = \frac{p}{q}$ with $p \in \mathbb{Z}$ and $q \in \mathbb{N}$ we define $x^{\alpha}=(x^{p})^{1/q}$ ### Properties - **Equivalent definition for rational power**: Using [[Inequalities for powers of positive reals with natural number exponents]], $(x^{p})^{\frac{1}{q}} = (x^{1/q})^{p}\iff [(x^{p})^{1/q}]^{q}=[(x^{1/q})^{p}]^{q}$ so we must check that the second equality holds. We have $[(x^{1/q})^{p}]^{q}=[(x^{1/q})]^{pq}=[(x^{1/q})]^{qp}=x^{p}=(x^{1/q})^{p}\quad\square$. - **Rational powers are consistent for equivalent exponents**: Suppose $\alpha= \frac{p}{q} =\frac{r}{s}$. WTS $x^{\alpha}x^{\beta}=x^{\alpha+\beta}$. Note that $(x^{p})^{1/q}=(x^{r})^{1/s}\iff(x^{p})^{s}=(x^{r})^{q}\iff x^{ps}=x^{rq}$ by taking the $qs$ power of both sides and we have $ps=rq$ since $p/q=r/s$ so LHS=RHS as required. - **Rational powers obey usual laws of indices**: (1) Take $\alpha=p/q$ and $\beta=r/s$ then $\alpha+\beta=(ps+qr)/qs$. Now we want to show that $(x^{p})^{1/q}(x^r)^{1/s}=(x^{ps+pr})^{1/qs}$. Taking the $qs$ power of the LHS we get $[(x^{p})^{1/q}]^{qs}[(x^{r})^{1/s}]^{qs}=(x^{p})^{s}(x^{r})^{q}=x^{ps}x^{rq}=x^{p s+pr}$which is the $qs$ power of the RHS. (2) $\alpha\beta=pr/qs$. We want to show that $(x^\alpha)^{\beta}=x^{\alpha\beta}$ which using our definition means we need to show that $\{[(x^{p})^{1/q}]^{r}\}^{1/s}=(x^{pr})^{1/qs}.$If we take the $qs$ power of the LHS, since $y^{qs}= (y^s)^{q}$ we get $()\{[(x^{p})^{1/q}]^{r}\}^{1/s})^{qs}=\{[(x^{p})^{1/q}]^{r}\}^{q}=\{[(x^{p})^{1/q}]^{q}\}^{r}=(x^{p})^{r}=x^{pr}$which is the same as the $qs$ power of the RHS. - **WTS** for $\alpha \in \mathbb{Q}$, $(x^{1/\alpha})^\alpha=x$. Suppose $\alpha=\frac{p}{q}$ we have $\{[(x^{q})^{1/p}]^{p}\}^{1/q}=\{ x^1 \}^{1/q}=x.$