Question

How do you divide $\dfrac{2}{{8i}}$ ?

Answer 1

Hint: In this question we are asked to divide the expression which is of complex numbers, we will do this by multiplying and dividing the expression with the conjugate of the denominator, i.e, here it will be$ - 8i$, and further multiplication and simplification the expression we will get the required result. And we have to remember that ${i^2} = - 1$, then combine all like terms i.e., combine real numbers with real numbers and imaginary numbers with imaginary numbers.

Complete step by step solution:
Given expression is $\dfrac{2}{{8i}}$,
Now multiplying and dividing the expression with the conjugate of the denominator, so here the conjugate of the denominator $8i$ will be $ - 8i$, we get,
$ \Rightarrow \dfrac{2}{{8i}} \times \dfrac{{ - 8i}}{{ - 8i}}$,
Now multiplying we get,
$ \Rightarrow \dfrac{{ - 16i}}{{ - 64{i^2}}}$,
Now simplifying we get,
$ \Rightarrow \dfrac{{16i}}{{64{i^2}}}$,
Now simplifying by using the fact ${i^2} = - 1$, substituting the value in the expression, we get,
$ \Rightarrow \dfrac{{16i}}{{64\left( { - 1} \right)}}$,
Now simplifying by factoring the numerator and denominator we get,
$ \Rightarrow \dfrac{{2 \times 2 \times 2 \times 2 \times i}}{{ - 2 \times 2 \times 2 \times 2 \times 2 \times 2}}$,
Now simplifying we get,
$ \Rightarrow \dfrac{i}{{ - 4}}$,
So the simplified form of the given expression is $\dfrac{{ - i}}{4}$.

$\therefore $ The simplified form of the expression when we divide $\dfrac{2}{{8i}}$ will be equal to $\dfrac{{ - i}}{4}$.

Note: Complex numbers are combination of two types of numbers i.e., real numbers and imaginary numbers, and they are defined by the inclusion of $i$ term, whose value is defined by $i = \sqrt { - 1} $, the general form for a complex number is defined by,
$z = a + ib$, where $z$ is the complex number, $a$is any real number and $b$ is the imaginary part of the complex number, both of which can be positive or negative. The complex numbers cannot be marked on a number line.