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$\dfrac{{5 + 4i}}{{4 + 5i}}$

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Hint – Any complex number can be converted into the form of A+iB where A is the real part and B is the imaginary part. Use rationalization of the denominator in order to simplify the given complex number. Use of algebraic identities will help you get the desired form.

Complete step-by-step answer:

Given complex number is

$\dfrac{{5 + 4i}}{{4 + 5i}}$

Now we have to convert this in the form of (A + iB).

So, first rationalize the number by (4 – 5i), (i.e. multiply and divide by (4 - 5i) in the given complex number) we have,

$ \Rightarrow \dfrac{{5 + 4i}}{{4 + 5i}} \times \dfrac{{4 - 5i}}{{4 - 5i}}$

Now multiply the numerator and in denominator apply the rule $\left[ {\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}} \right]$

$ \Rightarrow \dfrac{{20 - 25i + 16i - 20{i^2}}}{{16 - 25{i^2}}}$

Now as we know in complex the value of $\left[ {{i^2} = - 1} \right]$ so, use this property in above equation we have,

$ \Rightarrow \dfrac{{20 - 25i + 16i - 20\left( { - 1} \right)}}{{16 - 25\left( { - 1} \right)}}$

Now simplify the above equation we have,

$ \Rightarrow \dfrac{{20 - 9i + 20}}{{16 + 25}} = \dfrac{{40 - 9i}}{{41}}$

$ \Rightarrow \dfrac{{5 + 4i}}{{4 + 5i}} = \dfrac{{40}}{{41}} - i\dfrac{9}{{41}}$

So this is the required form.

Hence this is the required answer.

Note – Whenever we face such types of problems the key concept is simply to rationalize and simplify the given denominator part in order to be able to segregate the real and imaginary part to obtain the A+iB form for the given complex number.

Complete step-by-step answer:

Given complex number is

$\dfrac{{5 + 4i}}{{4 + 5i}}$

Now we have to convert this in the form of (A + iB).

So, first rationalize the number by (4 – 5i), (i.e. multiply and divide by (4 - 5i) in the given complex number) we have,

$ \Rightarrow \dfrac{{5 + 4i}}{{4 + 5i}} \times \dfrac{{4 - 5i}}{{4 - 5i}}$

Now multiply the numerator and in denominator apply the rule $\left[ {\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}} \right]$

$ \Rightarrow \dfrac{{20 - 25i + 16i - 20{i^2}}}{{16 - 25{i^2}}}$

Now as we know in complex the value of $\left[ {{i^2} = - 1} \right]$ so, use this property in above equation we have,

$ \Rightarrow \dfrac{{20 - 25i + 16i - 20\left( { - 1} \right)}}{{16 - 25\left( { - 1} \right)}}$

Now simplify the above equation we have,

$ \Rightarrow \dfrac{{20 - 9i + 20}}{{16 + 25}} = \dfrac{{40 - 9i}}{{41}}$

$ \Rightarrow \dfrac{{5 + 4i}}{{4 + 5i}} = \dfrac{{40}}{{41}} - i\dfrac{9}{{41}}$

So this is the required form.

Hence this is the required answer.

Note – Whenever we face such types of problems the key concept is simply to rationalize and simplify the given denominator part in order to be able to segregate the real and imaginary part to obtain the A+iB form for the given complex number.

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