Show that: (i) cube root of √729/cube root of √1000 = cube root of √729/1000

Question

Show that:

(i) \(\frac{\sqrt[3]{729}}{\sqrt[3]{1000}}\) = \(\sqrt[3]{\frac}{729}{1000}\)

(ii) \(\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}\) = \(\sqrt[3]{\frac}{-512}{343}\)

Answer 1

(i) \(\frac{\sqrt[3]{729}}{\sqrt[3]{1000}}\) = \(\sqrt[3]{\frac{729}{1000}}\)

We have,

LHS = \(\frac{\sqrt[3]{729}}{\sqrt[3]{1000}}\) 

= \(\frac{\sqrt[3]{9\times9\times9}}{\sqrt[3]{10\times10\times10}}\)

= \(\frac{\sqrt[3]{9^3}}{\sqrt{10^3}}\)

 = \(\frac{9}{10}\)

RHS,

= \(\sqrt[3]{\frac{729}{1000}}\)

= \(\sqrt[3]{\frac{9\times9\times9}{10\times10\times10}}\)

= \(\sqrt[3]{\frac{9^3}{10^3}}\)

= \(\frac{\sqrt[3]{9^3}}{\sqrt[3]{10^3}}\)

= \(\frac{9}{10}\)

∵ LHS = RHS

Hence, 

equation is true.

(ii) \(\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}\) = \(\sqrt[3]{\frac}{-512}{343}\)

LHS = \(\frac{\sqrt[3]{-512}}{\sqrt[3]{343}}\)

RHS, 

∵  LHS = RHS

Hence, 

equation is true.