Question

How do you simplify $8i+5\left( 4-1i \right)$ ?

Answer 1

Hint: The problem that we are given is based on the concept of complex numbers. In mathematics, a complex number is basically a number which we can express in the form of $x+yi$ , where $x$ and $y$ are real numbers and $i$ is a symbol known as the imaginary unit which is also satisfies the equation ${{i}^{2}}=-1$ . In the given expression we can eliminate the bracket with the help of the distributive law and get an expression containing some real numbers and imaginary numbers which must be added accordingly to get the simplified form of the expression.

Complete step by step solution:
The expression that we are given is
 $8i+5\left( 4-1i \right)$
The above expression consists of some real terms and terms having complex numbers. Hence, to get a further simplified form we must cancel the bracket with the help of the distributive law as shown below
$=8i+5\cdot 4-5\cdot 1i$
We multiply the terms $5$ and $4$ as
$=8i+20-5\cdot 1i$
Again, we multiply the terms $5$ and $1i$ as
$=8i+20-5i$
The above expression consists of two types of parts among which one is the real number part $20$ and the other is the imaginary number part $8i$ and $-5i$ . To further simplify the above expression, we can only add the like terms. Hence, we can only add an imaginary term with an imaginary term only. In the above expression we add the terms $8i$ and $5i$ as shown below
$=20+\left( 8i-5i \right)$
Further simplifying we get
$=20+3i$

Therefore, we conclude that the simplified form of the given expression is $20+3i$.

Note: While simplifying the given expression we have to add the like terms we must be very careful about the signs before the numbers present to avoid unnecessary mistakes. Also, we must not get confused between the real and imaginary terms, as people often get confused between the terms having $i$ and real numbers.