# Chapter 5 - Number Theory and the Real Number System - 5.3 The Rational Numbers - Exercise Set 5.3 - Page 285: 115

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 $-\frac{289}{{{2}^{4}}\cdot {{5}^{4}}\cdot 7}$.

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