Find the cube-roots of $\dfrac{343}{512}$

Prime factors of 343:

Prime factors of 512:

$\sqrt[3]{\dfrac{343}{512}}\\[1em] = {\dfrac{\sqrt[3]{343}}{\sqrt[3]{512}}}\\[1em] = {\dfrac{\sqrt[3]{7 \times 7 \times 7}}{\sqrt[3]{(2 \times 2 \times 2)\times(2 \times 2 \times 2)\times(2 \times 2 \times 2)}}}\\[1em] = {\dfrac{7}{2\times 2 \times 2}}\\[1em] = {\dfrac{7}{8}}$

Hence, $\sqrt[3]{\dfrac{343}{512}} = {\dfrac{7}{8}}$