Evaluate:

$(5^{-1} \times 3^{-1}) ÷ 6^{-1}$

As we know, for any non-zero rational number a

$a^{-n} = \dfrac{1}{a^n}$ and $a^{n} = \dfrac{1}{a^{-n}}$.

$(5^{-1} \times 3^{-1}) ÷ 6^{-1}\\[1em] = \Big(\dfrac{1}{5}^1 \times \dfrac{1}{3}^1\Big) ÷ \dfrac{1}{6}^1\\[1em] = \Big(\dfrac{1 \times 1}{5 \times 3}\Big) ÷ \dfrac{1}{6}^1\\[1em] = \Big(\dfrac{1}{15}\Big) ÷ \dfrac{1}{6}\\[1em] = \Big(\dfrac{1}{15}\Big) \times \dfrac{6}{1}\\[1em] = \Big(\dfrac{1 \times 6}{15 \times 1}\Big)\\[1em] = \dfrac{6}{15}\\[1em] = \dfrac{2}{5}$

Hence, $(5^{-1} \times 3^{-1}) ÷ 6^{-1} = \dfrac{2}{5}$