Question

Find The cube root of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that 20346417 = 9261 × 2197 .

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

 \[\sqrt[3]{20346417}\]

\[ = \sqrt[3]{9261 \times 2197}\]

\[= \sqrt[3]{9261} \times \sqrt[3]{2197}\] (By the above property)

\[= \sqrt[3]{3 \times 3 \times 3 \times 7 \times 7 \times 7} \times \sqrt[3]{13 \times 13 \times 13}\] (By prime factorisation)

\[= \sqrt[3]{\left\{ 3 \times 3 \times 3 \right\} \times \left\{ 7 \times 7 \times 7 \right\}} \times \sqrt[3]{\left\{ 13 \times 13 \times 13 \right\}}\]

\[ = 3 \times 7 \times 13\]

\[ = 273\]

Thus, the answer is 273.