Question

Find which of the following numbers are cubes of rational numbers:
(i) $\frac{27}{64}$
(ii) $\frac{125}{128}$
(iii) 0.001331
(iv) 0.04

Answer 1

(i)
We have:

$\frac{27}{64}=\frac{3\times 3\times 3}{8\times 8\times 8}=\frac{3^3}{8^3}={\left({\displaystyle \frac{3}{8}}\right)}^3$

Therefore, $\frac{27}{64}$ is a cube of $\frac{3}{8}$.

(ii)
We have:

$\frac{125}{128}=\frac{5\times 5\times 5}{2\times 2\times 2\times 2\times 2\times 2\times 2}=\frac{5^3}{2^3\times 2^3\times 2}$

It is evident that 128 cannot be grouped into triples of equal factors; therefore, $\frac{125}{128}$ is not a cube of a rational number.

(iii)
We have:

$0.001331=\frac{1331}{1000000}=\frac{11\times 11\times 11}{2\times 2\times 2\times 2\times 2\times 2\times 5\times 5\times 5\times 5\times 5\times 5}=\frac{{11}^3}{{\left(2\times 2\times 5\times 5\right)}^3}=\frac{{11}^3}{{100}^3}={\left({\displaystyle \frac{11}{100}}\right)}^3$

Therefore, $0.001331$ is a cube of $\frac{11}{100}$.

(iv)
We have:

$0.04=\frac{4}{100}=\frac{2\times 2}{2\times 2\times 5\times 5}$

It is evident that 4 and 100 could not be grouped in to triples of equal factors; therefore, 0.04 is not a cube of a rational number.