How do you simplify $\dfrac{{2 + 5i}}{{5 + 2i}}$ and write the complex number in standard form?

Hint: As we know that, the standard complex number is in the form of $x + iy$ , therefore we need to convert $\dfrac{{2 + 5i}}{{5 + 2i}}$ , in the standard form of a complex number. We will do that with the help of a rationalization method. In other words, we can say that we rationalize a fraction by multiplying and dividing the fraction with the conjugate of the denominator.

Formula used:
Conjugate of a complex number: Let $z = x + iy$ , then $\bar z = x - iy$

Complete step-by-step answer:
First of all, in order to convert $\dfrac{{2 + 5i}}{{5 + 2i}}$ , in the standard form of a complex number, which is the real part and the imaginary part, we need to rationalize $\dfrac{{2 + 5i}}{{5 + 2i}}$ .
Now, in order to rationalize the above fraction, let us first find the conjugate of the denominator. We find the conjugate of a complex number by changing the sign of the imaginary part.
So we get, $\overline {5 + 2i} = 5 - 2i$ .
Now, first, let us take $z = \dfrac{{2 + 5i}}{{5 + 2i}}$
On rationalizing, we get
$\Rightarrow$$z = \dfrac{{2 + 5i}}{{5 + 2i}} \times \dfrac{{5 - 2i}}{{5 - 2i}}$
This becomes as
$\Rightarrow$$z = \dfrac{{(2 + 5i)(5 - 2i)}}{{(5 + 2i)(5 - 2i)}}$
By simplifying the brackets in numerator and denominator, we get
$\Rightarrow$$z = \dfrac{{2(5 - 2i) + 5i(5 - 2i)}}{{{5^2} - {{\left( {2i} \right)}^2}}}$
In the denominator, we use $(a + b)(a - b) = {a^2} - {b^2}$
$ \Rightarrow \dfrac{{10 - 4i + 25i - 10{i^2}}}{{25 - 4{i^2}}}$
Here we use ${i^2} = - 1$ and simplify the numerator and denominator using the BODMAS rule.
$ \Rightarrow \dfrac{{10 + 21i - 10( - 1)}}{{25 - 4( - 1)}}$
$ \Rightarrow \dfrac{{10 + 21i + 10}}{{25 + 4}}$
$ \Rightarrow \dfrac{{20 + 21i}}{{29}}$
Now, in order to convert the above fraction into the standard form of a complex number, we will write the real part and the imaginary part separately.
So, this can also be written as
$\Rightarrow$$\dfrac{{20}}{{29}} + \dfrac{{21}}{{29}}i$ .
Hence, we get $z = \dfrac{{20}}{{29}} + \dfrac{{21}}{{29}}i$, which is in the standard form of a complex number.

Here, $\operatorname{Re} (z) = \dfrac{{20}}{{29}}$ and $\operatorname{Im} (z) = \dfrac{{21}}{{29}}$

Note:
$Re(z)$ stands for the real part of the complex number which is denoted as $z$ and $\operatorname{Im} (z)$ stands for the imaginary part of the complex number which is denoted as $z$ . We often take $z = x + iy$ as the standard form of a complex number, where $\operatorname{Re} (z) = x$ and $\operatorname{Im} (z) = y$