Question

Find the cube root of the following rational number 0.001728 .

Answer 1

We have:

\[0 . 001728 = \frac{1728}{1000000}\]

∴ \[\sqrt[3]{0 . 001728} = \sqrt[3]{\frac{1728}{1000000}} = \frac{\sqrt[3]{1728}}{\sqrt[3]{1000000}}\]

Now
On factorising 1728 into prime factors, we get:

\[1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3\]

On grouping the factors in triples of equal factors, we get:

\[1728 = \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 2 \times 2 \times 2 \right\} \times \left\{ 3 \times 3 \times 3 \right\}\]

Now, taking one factor from each triple, we get:

\[\sqrt[3]{1728} = 2 \times 2 \times 3 = 12\]

Also

\[\sqrt[3]{1000000} = \sqrt[3]{100 \times 100 \times 100} = 100\]

∴ \[\sqrt[3]{0 . 001728} = \frac{\sqrt[3]{1728}}{\sqrt[3]{1000000}} = \frac{12}{100} = 0 . 12\]