Calculations with permutations: show that $(1,2,3)^2(5,7)^2=(1,3,2)$ How can I show $(1,2,3)^2(5,7)^2=(1,3,2)$? And, specifically, what does $(1,2,3)^2$ and $(5,7)^2$ equal individually?

Note that with $(5, 7)^2 = (5, 7)(5, 7)$, $\quad (5 \to 7 \to 5)$ and $(7 \to 5 \to 7),\quad $ which gives us $(5, 7)(5, 7) = (1)$, the identity permutation. Any two-cycle, squared, gives us the identity permutation: it's an order two permutation.
Since the first two squared cycles are disjoint from one another, you can simply compute $$(1, 2, 3)^2 \cdot (5, 7)^2 = (1, 2, 3)(1, 2, 3)\cdot(1) = (1, 2, 3)(1, 2, 3) = (1, 3, 2)$$
