Question

Evaluate: 

\[\sqrt[3]{100} \times \sqrt[3]{270}\]

 

Answer 1

100 and 270 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

\[\therefore \sqrt[3]{100} \times \sqrt[3]{270}\]

\[ = \sqrt[3]{100 \times 270}\]

\[= \sqrt[3]{\left( 2 \times 2 \times 5 \times 5 \right) \times \left( 2 \times 3 \times 3 \times 3 \times 5 \right)}\]                 (By prime factorisation)

\[= \sqrt[3]{\left\{ 2 \times 2 \times 2 \right\} \times \left\{ 3 \times 3 \times 3 \right\} \times \left\{ 5 \times 5 \times 5 \right\}}\]

\[ = 2 \times 3 \times 5\]

\[ = 30\]

Thus, the answer is 30.