Question

Show that: \[\sqrt[3]{- 125 \times 216} = \sqrt[3]{- 125} \times \sqrt[3]{216}\]

Answer 1

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.