Question

Find The cube root of the numbers 3048625, 20346417, 210644875, 57066625 using the fact that 3048625 = 3375 × 729 .

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

\[\sqrt[3]{3048625}\]

\[ = \sqrt[3]{3375 \times 729}\]

\[= \sqrt[3]{3375} \times \sqrt[3]{729}\]     (By the above property)

 

\[= \sqrt[3]{3 \times 3 \times 3 \times 5 \times 5 \times 5} \times \sqrt[3]{9 \times 9 \times 9}\]      (By prime factorisation)

\[= \sqrt[3]{\left\{ 3 \times 3 \times 3 \right\} \times \left\{ 5 \times 5 \times 5 \right\}} \times \sqrt[3]{\left\{ 9 \times 9 \times 9 \right\}}\]
\[ = 3 \times 5 \times 9\]
\[ = 135\]

Thus, the answer is 135.