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# How do you simplify $\dfrac{{7 + 2i}}{{4 + 5i}}?$

Last updated date: 18th Sep 2024
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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.