Question

Find the cube root of the decimal: $19.683$

Answer 1

Step 1: Convert the decimal into a fraction and find factors

The given decimal is $19.683$.

Here, $19.683=\frac{19683}{1000}$

Now, calculate the prime factors of $19683$ and $1000$ as,

So, $19683=3\times 3\times 3\times 3\times 3\times 3\times 3\times 3\times 3$

And, $1000=2\times 2\times 2\times 5\times 5\times 5$

Step 2: Calculation of the cube root

Then, the cube root of $19.683$ is,

$\sqrt[3]{19.683}=\sqrt[3]{\frac{19683}{1000}}$

$\Rightarrow \sqrt[3]{19.683}=\sqrt[3]{\frac{3\times 3\times 3\times 3\times 3\times 3\times 3\times 3\times 3}{2\times 2\times 2\times 5\times 5\times 5}}$

$\Rightarrow \sqrt[3]{19.683}=\frac{\sqrt[3]{3\times 3\times 3\times 3\times 3\times 3\times 3\times 3\times 3}}{\sqrt[3]{2\times 2\times 2\times 5\times 5\times 5}}$ $\left[\because \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\right]$

$$\begin{gathered} \Rightarrow \sqrt[3]{19.683}=\frac{\sqrt[3]{\overline{3\times 3\times 3}\times \overline{3\times 3\times 3}\times \overline{3\times 3\times 3}}}{\sqrt[3]{\overline{2\times 2\times 2}\times \overline{5\times 5\times 5}}} \\ \Rightarrow \sqrt[3]{19.683}=\frac{3\times 3\times 3}{2\times 5} \\ \Rightarrow \sqrt[3]{19.683}=\frac{27}{10} \\ \Rightarrow \sqrt[3]{19.683}=2.7 \end{gathered}$$

Hence, the cube root of $19.683$ is $2.7$