Alg3.3.13

Simplify $[\frac{a^p}{a^q}]^{p+q}\cdot [\frac{a^q}{a^r}]^{q+r}\cdot [\frac{a^r}{a^p}]^{r+p}\,$

Given $[\frac{a^p}{a^q}]^{p+q}\cdot [\frac{a^q}{a^r}]^{q+r}\cdot [\frac{a^r}{a^p}]^{r+p}\,$

Simplifying we have

$(a^{p-q})^{p+q}\cdot (a^{q-r})^{q+r} \cdot (a^{r-p})^{r+p}\,$

$a^{(p+q)(p-q)} \cdot a^{(q+r)(q-r)} \cdot a^{(r+p)(r-p)}\,$

$a^{p^2-q^2} \cdot a^{q^2-r^2} \cdot a^{r^2-p^2}\,$

Since bases of the three are equal,

$a^{p^2-q^2+q^2-r^2+r^2-p^2}=a^{0}=1\,$

Hence the value of the expression after simplification is 1.