Answer
The solution of the given operation is \[-\frac{289}{{{2}^{4}}\cdot {{5}^{4}}\cdot 7}\].
Work Step by Step
Two rational numbers \[\frac{a}{b}\]and \[\frac{c}{d}\] can be added or 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 \[{{2}^{4}}\cdot {{5}^{4}}\cdot 7\].
The given operation can be performed as follows:
\[\begin{align}
& \frac{1}{{{2}^{4}}\cdot {{5}^{3}}\cdot 7}+\frac{1}{2\cdot {{5}^{4}}}-\frac{1}{{{2}^{3}}\cdot {{5}^{2}}}=\frac{1}{{{2}^{4}}\cdot {{5}^{3}}\cdot 7}\times \frac{5}{5}+\frac{1}{2\cdot {{5}^{4}}}\times \frac{{{2}^{3}}\cdot 7}{{{2}^{3}}\cdot 7}-\frac{1}{{{2}^{3}}\cdot {{5}^{2}}}\times \frac{2\cdot {{5}^{2}}\cdot 7}{2\cdot {{5}^{2}}\cdot 7} \\
& =\frac{5}{{{2}^{4}}\cdot {{5}^{4}}\cdot 7}+\frac{56}{{{2}^{4}}\cdot {{5}^{4}}\cdot 7}-\frac{350}{{{2}^{4}}\cdot {{5}^{4}}\cdot 7} \\
& =\frac{5+56-350}{{{2}^{4}}\cdot {{5}^{4}}\cdot 7} \\
& =-\frac{289}{{{2}^{4}}\cdot {{5}^{4}}\cdot 7}
\end{align}\]
The solution of the given operation is \[-\frac{289}{{{2}^{4}}\cdot {{5}^{4}}\cdot 7}\].
