Question

Find The cube root of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that  210644875 = 42875 × 4913 .

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

\[\sqrt[3]{210644875}\]

\[ = \sqrt[3]{42875 \times 4913}\]

\[= \sqrt[3]{42875} \times \sqrt[3]{4913}\]   (By the above property)

\[= \sqrt[3]{5 \times 5 \times 5 \times 7 \times 7 \times 7} \times \sqrt[3]{17 \times 17 \times 17}\]      (By prime factorisation)

\[= \sqrt[3]{\left\{ 5 \times 5 \times 5 \right\} \times \left\{ 7 \times 7 \times 7 \right\}} \times \sqrt[3]{\left\{ 17 \times 17 \times 17 \right\}}\]
\[ = 5 \times 7 \times 17\]
\[ = 595\]

Thus, the answer is 595.