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# How do you simplify the given term of complex numbers, the term is$\dfrac{{5 - i}}{{5 + i}}$ ?

Last updated date: 13th Sep 2024
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Hint:Complex term can be treated as algebraic variables, but the thing is the properties used here are a bit different, all calculations can be done like for variables, in the final answer only you can have to put the predefined values of complex numbers, rest everything is same and if applicable certain properties are needed to be used.

Formulae Used:
${a^2} - {b^2} = (a + b)(a - b)$

The given complex term is $\dfrac{{5 - i}}{{5 + i}}$. Here we have to multiply the term in numerator and denominator with the conjugate of denominator such that we can proceed further to obtain the best possible solution for the question, on solving we get:
$\dfrac{{5 - i}}{{5 + i}} \times \dfrac{{5 - i}}{{5 - i}} \\ \Rightarrow \dfrac{{{{(5 - i)}^2}}}{{{5^2} - {i^2}}}\left( {using\,the\,formulae\,{a^2} - {b^2} = (a + b)(a - b)} \right) \\ \Rightarrow \dfrac{{{5^2} + {i^2} - 10i}}{{25 - ( - 1)}} \\ \Rightarrow \dfrac{{25 + ( - 1) - 10i}}{{26}} \\ \Rightarrow \dfrac{{25 - 10i}}{{26}} \\ \therefore \dfrac{{25}}{{26}} - \dfrac{{10}}{{26}}i$