Question

How do you write the following quotient in standard form \[\dfrac{6-7i}{i}\] ?

Answer 1

Hint: To solve this first we have to get rid of ‘i’ in the denominator. So we will multiply the denominator and numerator with denominators conjugate. After multiplying we have to simplify the equation accordingly to get the standard form.

Complete step-by-step answer:
Lets know about the conjugate of the given complex number.
If the complex number given is in the form of \[a+ib\]. We have to change the sign of the imaginary part to get the conjugate. Then we will get \[a-ib\].
The product of a complex number and its conjugate is a real number: \[{{a}^{2}}+{{b}^{2}}\]\[\]
If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root.

Given quotient is
\[\dfrac{6-7i}{i}\]
We can write the denominator as \[0+i\]. So its conjugate is negative sign of imaginary part.
We will get the conjugate as \[0-i\] .
So now we have to multiply and divide the quotient with the conjugate of denominator \[0-i\]
We will get
\[\Rightarrow \dfrac{6-7i}{0+i}.\dfrac{0-i}{0-i}\]
Now we have to multiply the two terms.
We will get
\[\Rightarrow \dfrac{-6i+7{{i}^{2}}}{-{{i}^{2}}}\]
We already know that \[{{i}^{2}}=-1\] . so we can substitute this in the denominator and numerator.
We will get
\[\Rightarrow \dfrac{-6i+7\left( -1 \right)}{-\left( -1 \right)}\]
By simplifying it we will get
\[\Rightarrow \dfrac{-7-6i}{1}\]
\[\Rightarrow -7-6i\]
The standard form that we can get from the quotient \[\dfrac{6-7i}{i}\] is \[-7-6i\].

Note: We should be careful while eliminating i in the denominator to apply a correct formula. Otherwise there is a chance of creating new complex terms instead eliminating them. We should be aware of important complex numbers formulas to solve these types of equations.