Question

Find The cube root of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that 57066625 = 166375 × 343 .

Answer 1

To find the cube root, we use the following property:

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

Now

\[\sqrt[3]{57066625}\]
\[ = \sqrt[3]{166375 \times 343}\]

\[= \sqrt[3]{166375} \times \sqrt[3]{343}\]

 (By the above property)

\[= \sqrt[3]{5 \times 5 \times 5 \times 11 \times 11 \times 11} \times \sqrt[3]{7 \times 7 \times 7}\]   (By prime factorisation)

\[= \sqrt[3]{\left\{ 5 \times 5 \times 5 \right\} \times \left\{ 11 \times 11 \times 11 \right\}} \times \sqrt[3]{\left\{ 7 \times 7 \times 7 \right\}}\]

\[ = 5 \times 11 \times 7\]

\[ = 385\]

Thus, the answer is 385.