How do you simplify \[\dfrac{{7 + 2i}}{{4 + 5i}}?\]

Hint: We will multiply numerator and denominator by the complement complex number of \[4 + 5i\] and then simplify the above iteration. Finally we get the required answer.

Complete Step by Step Solution:
The given expression is \[\dfrac{{7 + 2i}}{{4 + 5i}}.\]
Now, we will multiply numerator and denominator by \[(4 - 5i)\].
By doing it, we get:
\[ \Rightarrow \dfrac{{(7 + 2i) \times (4 - 5i)}}{{(4 + 5i) \times (4 - 5i)}}\].
Now, by using the formula, we can write the denominator as following way:
\[ \Rightarrow \dfrac{{(7 + 2i) \times (4 - 5i)}}{{{{(4)}^2} - {{(5i)}^2}}}\].
Now, by doing further simplification, we get:
\[ \Rightarrow \dfrac{{(7 + 2i) \times (4 - 5i)}}{{16 - (25 \times - 1)}},\;as\;{i^2} = - 1.\]
By doing further simplification:
\[ \Rightarrow \dfrac{{(7 + 2i) \times (4 - 5i)}}{{16 + 25}}\]
\[ \Rightarrow \dfrac{{(7 + 2i) \times (4 - 5i)}}{{41}}....................(1)\]
Now, calculate the numerator part only, we get:
\[ \Rightarrow (7 + 2i) \times (4 - 5i)\]
\[ \Rightarrow (7 + 2i) \times 4 - (7 + 2i) \times 5i\].
Using multiplication, we get:
\[ \Rightarrow (28 + 8i) - (35i + 10{i^2})\].
Now, using algebraic calculations and putting the value of \[{i^2} = - 1\], we get:
\[ \Rightarrow (28 + 8i) - (35i - 10)\]
\[ \Rightarrow 28 + 8i - 35i - 10\].
Now, by doing further simplification:
\[ \Rightarrow (18 - 27i)\].
Now, putting the value of numerator of \[(18 - 27i)\] in the iteration \[(1)\], we get:
\[ \Rightarrow \dfrac{{(18 - 27i)}}{{41}}\].

Therefore, the required answer is \[\dfrac{{(18 - 27i)}}{{41}}\].

Note: Points to remember:
A complex number is expressed as following:
\[X + i.Y\], where \[X\] and \[Y\] are real numbers but the imaginary part of the number is \[i\].
A complex number lies on the imaginary axis in \[X - Y\] plane.