Question

Find the cube root of each of the following rational numbers:
(i) 0.001728
(ii) 0.003375
(iii) 0.001
(iv) 1.331

Answer 1

(i)
We have:

$0.001728=\frac{1728}{1000000}$

$\therefore$ $\sqrt[3]{0.001728}=\sqrt[3]{\frac{1728}{1000000}}=\frac{\sqrt[3]{1728}}{\sqrt[3]{1000000}}$

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

$\therefore$ $\sqrt[3]{0.001728}=\frac{\sqrt[3]{1728}}{\sqrt[3]{1000000}}=\frac{12}{100}=0.12$

(ii)
We have:

$0.003375=\frac{3375}{1000000}$

$\therefore$ $\sqrt[3]{0.003375}=\sqrt[3]{\frac{3375}{1000000}}=\frac{\sqrt[3]{3375}}{\sqrt[3]{1000000}}$

Now

On factorising 3375 into prime factors, we get:
$3375=3\times 3\times 3\times 5\times 5\times 5$

On grouping the factors in triples of equal factors, we get:
$3375=\left\{3\times 3\times 3\right\}\times \left\{5\times 5\times 5\right\}$

Now, taking one factor from each triple, we get:
$\sqrt[3]{3375}=3\times 5=15$

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

$\therefore$ $\sqrt[3]{0.003375}=\frac{\sqrt[3]{3375}}{\sqrt[3]{1000000}}=\frac{15}{100}=0.15$

(iii)
We have:

$0.001=\frac{1}{1000}$

$\therefore$ $\sqrt[3]{0.001}=\sqrt[3]{\frac{1}{1000}}=\frac{\sqrt[3]{1}}{\sqrt[3]{1000}}=\frac{1}{10}=0.1$
(iv)
We have:

$1.331=\frac{1331}{1000}$

$\therefore$ $\sqrt[3]{1.331}=\sqrt[3]{\frac{1331}{1000}}=\frac{\sqrt[3]{1331}}{\sqrt[3]{1000}}=\frac{\sqrt[3]{11\times 11\times 11}}{\sqrt[3]{1000}}=\frac{11}{10}=1.1$