Answer
Two rational numbers \[\frac{a}{b}\]and \[\frac{c}{d}\] can be subtracted by first finding the least common multiple of their denominators also known as least common denominator. The rational numbers are then multiplied by a rational number of the form \[\frac{e}{e}\], so that the denominator of both the rational numbers becomes the least common denominators as found earlier.
The least common denominator of the given rational numbers is \[{{5}^{3}}\cdot {{3}^{2}}\].
The given operation can be performed as follows:
\[\begin{align}
& \frac{7}{{{5}^{2}}\cdot {{3}^{2}}}-\frac{1}{3\cdot {{5}^{3}}}=\frac{7}{{{5}^{2}}\cdot {{3}^{2}}}\times \frac{5}{5}-\frac{1}{3\cdot {{5}^{3}}}\times \frac{3}{3} \\
& =\frac{35}{{{5}^{3}}\cdot {{3}^{2}}}-\frac{3}{{{3}^{2}}\cdot {{5}^{3}}} \\
& =\frac{35-3}{{{5}^{3}}\cdot {{3}^{2}}} \\
& =\frac{32}{{{5}^{3}}\cdot {{3}^{2}}}
\end{align}\]
This is further simplified as,
\[\frac{3}{{{2}^{2}}\cdot {{3}^{2}}}=\frac{1}{{{2}^{2}}\cdot 3}\]
So, the solution of the given operation is \[\frac{32}{{{5}^{3}}\cdot {{3}^{2}}}\].
