Question

Evaluate:

\[\sqrt[3]{36} \times \sqrt[3]{384}\]

 

Answer 1

36 and 384 are not perfect cubes; therefore, we use the following property:

\[\sqrt[3]{ab} = \sqrt[3]{a} \times \sqrt[3]{b}\]   for any two integers a and b

\[\therefore \sqrt[3]{36} \times \sqrt[3]{384}\]

\[ = \sqrt[3]{36 \times 384}\]

\[= \sqrt[3]{\left( 2 \times 2 \times 3 \times 3 \right) \times \left( 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \right)}\]        (By prime factorisation)

\[= \sqrt[3]{\left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 3 \times 3 \times 3 \right\}}\]

\[ = 2 \times 2 \times 2 \times 3\]

\[ = 24\]

Thus, the answer is 24.