Answer
The solution of the given operation is \[\frac{29}{{{2}^{3}}\cdot 3\cdot {{17}^{9}}}\].
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 denominator as found earlier.
The least common denominator of the given rational numbers is \[{{2}^{3}}\cdot 3\cdot {{17}^{9}}\].
The given operation can be performed as follows:
\[\begin{align}
& \frac{1}{{{2}^{3}}\cdot {{17}^{8}}}+\frac{1}{2\cdot {{17}^{9}}}-\frac{1}{{{2}^{2}}\cdot 3\cdot {{17}^{8}}}=\frac{1}{{{2}^{3}}\cdot {{17}^{8}}}\times \frac{3\cdot 17}{3\cdot 17}+\frac{1}{2\cdot {{17}^{9}}}\times \frac{3\cdot {{2}^{2}}}{3\cdot {{2}^{2}}}-\frac{1}{{{2}^{2}}\cdot 3\cdot {{17}^{8}}}\times \frac{2\cdot 17}{2\cdot 17} \\
& =\frac{51}{{{2}^{3}}\cdot 3\cdot {{17}^{9}}}+\frac{12}{{{2}^{3}}\cdot 3\cdot {{17}^{9}}}-\frac{34}{{{2}^{3}}\cdot 3\cdot {{17}^{9}}} \\
& =\frac{51+12-34}{{{2}^{3}}\cdot 3\cdot {{17}^{9}}} \\
& =\frac{29}{{{2}^{3}}\cdot 3\cdot {{17}^{9}}}
\end{align}\]
The solution of the given operation is \[\frac{29}{{{2}^{3}}\cdot 3\cdot {{17}^{9}}}\].
