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

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Last updated date: 27th Feb 2024
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IVSAT 2024
Answer
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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:
\[{a^2} - {b^2} = (a + b)(a - b)\]

Complete step by step answer:
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:
\[\dfrac{{5 - i}}{{5 + i}} \times \dfrac{{5 - i}}{{5 - i}} \\
\Rightarrow \dfrac{{{{(5 - i)}^2}}}{{{5^2} - {i^2}}}\left( {using\,the\,formulae\,{a^2} - {b^2} = (a + b)(a - b)} \right) \\
\Rightarrow \dfrac{{{5^2} + {i^2} - 10i}}{{25 - ( - 1)}} \\
\Rightarrow \dfrac{{25 + ( - 1) - 10i}}{{26}} \\
\Rightarrow \dfrac{{25 - 10i}}{{26}} \\
\therefore \dfrac{{25}}{{26}} - \dfrac{{10}}{{26}}i \]
This is the final required solution for the given term, since here no further simplification can be done hence it is the simplest possible answer.

Additional Information:
Here we can multiply and divide with the conjugate of numerator also but by using that we can’t separate the real and complex term into two fractions, as we did. Because the complex part will come under a denominator and then separation can’t be done.

Note: For the question in which simplification is needed, then for such questions you need to think that how you can obtain the simplest form of the equation, like for here we have to separate the real and complex term, so accordingly we think about the steps and get the required best possible solution for the question.