Question

How do you write ${\left( {2 + 3i} \right)^2}$ in standard form?

Answer 1

Hint:We know that a complex number has two components. The first part is a real number and the second part is an imaginary part. For example, let us take $x$ is the real part and $y$is the imaginary part of a complex number. Then, this complex number can be written in the standard form as $x + yi$. Thus, we will convert the given complex number into this type of standard form.

Formula used:
${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$

Complete step by step answer:
Our first step is to expand the given complex number ${\left( {2 + 3i} \right)^2}$.We have the formula for expanding a perfect square ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.For this question, we will take $a = 2$ and $b = 3i$.
$
{\left( {2 + 3i} \right)^2} \\
\Rightarrow {\left( 2 \right)^2} + \left( {2 \times 2 \times 3i} \right) + {\left( {3i} \right)^2} \\
\Rightarrow 4 + 12i + 9{i^2} \\ $
We know that the value of ${i^2}$is $ - 1$.
$4 + 12i + 9\left( { - 1} \right) \\
\Rightarrow 4 + 12i - 9 \\
\therefore - 5 + 12i $
Here, $ - 5$ is the real part of this standard form of complex number and $12$ is the imaginary part of this standard form of complex number.

Hence, our answer is: ${\left( {2 + 3i} \right)^2}$is written in its standard form as $ - 5 + 12i$.

Note:In this question, we have seen that the complex number is a combination of a real number and an imaginary number. Any numbers which are present in a number system such as positive, negative, zero, integer, rational, irrational, fractions are real numbers. Whereas, the numbers which are not real are imaginary numbers. When we square an imaginary number, it gives a negative result. The main application of these complex numbers is to represent periodic motions such as water waves, alternating current, light waves and other such functions which rely on sine or cosine waves.