Find which of the following numbers are cubes of rational numbers:

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

Solution:

(i)
We have:

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

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:

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

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.