Question

How do you divide \[\dfrac{1}{-8-5i}\]?

Answer 1

Hint: This question is from the topic of pre-calculus. In this question, we will first understand how to find the complex conjugate of any complex number. After that, we will understand how to rationalize a complex number. After that, we will solve the term \[\dfrac{1}{-8-5i}\]. We will first rationalize the term \[\dfrac{1}{-8-5i}\] and then solve the further process. After that, we will get our answer.

Complete step by step answer:
Let us solve this question.
In this question, we have asked to divide the term which is given in the question. The term which we have to divide is \[\dfrac{1}{-8-5i}\]. We can say that we have to solve the term \[\dfrac{1}{-8-5i}\].
So, before solving, let us first understand the complex conjugate of a complex number.
Let us understand this from an example of complex number that is \[a+ib\]
The complex conjugate of \[a+ib\] will be \[a-ib\].
Now, let us understand about rationalization.
Whenever we have to rationalize a complex number, we will multiply the complex conjugate of that complex number to the numerator and denominator.
The given term which we have to solve is
\[\dfrac{1}{-8-5i}\]
The number \[-8-5i\] is a complex number.
Let us rationalize the term with complex conjugate of the complex number \[-8-5i\]
We can write the term as \[\dfrac{1}{-8-5i}\] as
\[\dfrac{1}{-8-5i}=\dfrac{1}{-8-5i}\times \dfrac{\left( -8+5i \right)}{\left( -8+5i \right)}\]
Using the formula \[{{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\], we can write the above equation as
\[\Rightarrow \dfrac{1}{-8-5i}=\dfrac{\left( -8+5i \right)}{{{\left( -8 \right)}^{2}}-{{\left( 5i \right)}^{2}}}\]
The above equation can also be written as
\[\Rightarrow \dfrac{1}{-8-5i}=\dfrac{\left( -8+5i \right)}{64-25{{\left( i \right)}^{2}}}\]
As we know that \[{{\left( i \right)}^{2}}=-1\], so we can write the above equation as
\[\Rightarrow \dfrac{1}{-8-5i}=\dfrac{\left( -8+5i \right)}{64-25\left( -1 \right)}\]
\[\Rightarrow \dfrac{1}{-8-5i}=\dfrac{\left( -8+5i \right)}{64+25}\]
\[\Rightarrow \dfrac{1}{-8-5i}=\dfrac{\left( -8+5i \right)}{89}\]
The above equation can also be written as
\[\Rightarrow \dfrac{1}{-8-5i}=-\dfrac{8}{89}+\dfrac{5i}{89}\]
Hence, we have divided the term \[\dfrac{1}{-8-5i}\] and got the answer as \[-\dfrac{8}{89}+\dfrac{5i}{89}\].

Note:
We should have a better knowledge in the topic of pre-calculus for solving this type of question easily. We should remember the following formulas for solving this type of question:
\[{{a}^{2}}-{{b}^{2}}=\left( a-b \right)\left( a+b \right)\]
 \[{{\left( i \right)}^{2}}=-1\]
We should remember how to rationalize the complex numbers.