Question

How do you simplify $4\left( {2 - 3i} \right) + 6i$?

Answer 1

This problem deals with simplifying the complex numbers. A complex number is a number that can be expressed in the form of $a + ib$, where $a$ and $b$ are real numbers, and $i$ represents the imaginary unit, satisfying the equation ${i^2} = - 1$. Because no real number satisfies this equation, $i$ is called an imaginary number.

Complete step by step answer:
The given expression is $4\left( {2 - 3i} \right) + 6i$, we have to simplify the expression.
Now consider the given expression, as shown below:
$\Rightarrow 4\left( {2 - 3i} \right) + 6i$
We have to simplify in such a way that, first simplifying the first term and then simplifying the second term. Then simplifying both the first and the second term, as shown below:
Now simplifying this expression by solving the expressions in the first and second terms.
Consider the first term as shown:
$\Rightarrow 4\left( {2 - 3i} \right)$
Multiplying the number 4, with each term in the bracket inside, as shown below:
$\Rightarrow 4\left( {2 - 3i} \right) = 8 - 12i$
Now consider the second term as shown below:
$\Rightarrow 6i$
Now adding both the first term and the second terms, as shown below:
$\Rightarrow 4\left( {2 - 3i} \right) + 6i$
$\Rightarrow 8 - 12i + 6i$
Here simplifying the terms$- 12i$ and $6i$ as shown below:
$\Rightarrow 8 - 6i$
Now taking the number 2 common from the first two terms, as shown below:
$\Rightarrow 8 - 6i = 2\left( {4 - 3i} \right)$
So the simplification of the given expression $4\left( {2 - 3i} \right) + 6i$ is equal to $8 - 6i$.
$\therefore 4\left( {2 - 3i} \right) + 6i = 8 - 6i$

The value of the given expression $4\left( {2 - 3i} \right) + 6i$ is $8 - 6i$.

Note: Please note that the backbone of this new number system is the number $i$, also known as the imaginary unit. So from here we can conclude that any imaginary number is also a complex number, and any real number is also a complex number.