Question

Show that:
(i) $\sqrt[3]{27}\times \sqrt[3]{64}=\sqrt[3]{27\times 64}$
(ii) $\sqrt[3]{64\times 729}=\sqrt[3]{64}\times \sqrt[3]{729}$
(iv) $\sqrt[3]{-125\times 216}=\sqrt[3]{-125}\times \sqrt[3]{216}$
(v) $\sqrt[3]{-125-1000}=\sqrt[3]{-125}\times \sqrt[3]{-1000}$

Answer 1

(i)
LHS = $\sqrt[3]{27}\times \sqrt[3]{64}=\sqrt[3]{3\times 3\times 3}\times \sqrt[3]{4\times 4\times 4}=3\times 4=12$

RHS = $\sqrt[3]{27\times 64}=\sqrt[3]{3\times 3\times 3\times 4\times 4\times 4}=\sqrt[3]{\left\{3\times 3\times 3\right\}\times \left\{4\times 4\times 4\right\}}=3\times 4=12$

Because LHS is equal to RHS, the equation is true.

(ii)
LHS = $\sqrt[3]{64\times 729}=\sqrt[3]{4\times 4\times 4\times 9\times 9\times 9}=\sqrt[3]{\left\{4\times 4\times 4\right\}\times \left\{9\times 9\times 9\right\}}=4\times 9=36$

RHS = $\sqrt[3]{64}\times \sqrt[3]{729}=\sqrt[3]{4\times 4\times 4}\times \sqrt[3]{9\times 9\times 9}=4\times 9=36$

Because LHS is equal to RHS, the equation is true.

(iii)
LHS = $\sqrt[3]{-125\times 216}=\sqrt[3]{-5\times -5\times -5\times \left\{2\times 2\times 2\times 3\times 3\times 3\right\}}=\sqrt[3]{\left\{-5\times -5\times -5\right\}\times \left\{2\times 2\times 2\right\}\times \left\{3\times 3\times 3\right\}}=-5\times 2\times 3=-30$

RHS = $\sqrt[3]{-125}\times \sqrt[3]{216}=\sqrt[3]{-5\times -5\times -5}\times \sqrt[3]{\left\{2\times 2\times 2\right\}\times \left\{3\times 3\times 3\right\}}=-5\times \left(2\times 3\right)=-30$

Because LHS is equal to RHS, the equation is true.

(iv)
LHS = $\sqrt[3]{-125\times -1000}=\sqrt[3]{-5\times -5\times -5\times -10\times -10\times -10}=\sqrt[3]{\left\{-5\times -5\times -5\right\}\times \left\{-10\times -10\times -10\right\}}=-5\times -10=50$

RHS = $\sqrt[3]{-125}\times \sqrt[3]{-1000}=\sqrt[3]{-5\times -5\times -5}\times \sqrt[3]{\left\{-10\times -10\times -10\right\}}=-5\times -10=50$

Because LHS is equal to RHS, the equation is true.