Question

How do you simplify \[\dfrac{1}{{{i^3}}}\] ?

Answer 1

Hint: Here in this question, we have to simplify the given number, the given number in the form of fraction and in the denominator we have a ‘ \[{i^3}\] ’ where i represent the imaginary number. By using the properties on imaginary numbers and knowing i raise to 4n is equal to 1 and on further simplification by using the value of i, we obtain the required solution for the question

Complete step-by-step answer:
Imaginary numbers are numbers that are not real or An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit I which is defined by its property. \[i = \sqrt { - 1} \] or \[{i^2} = - 1\] . The imaginary numbers are the numbers when squared it gives the negative result. In other words, imaginary numbers are defined as the square root of the negative numbers where it does not have a definite value, i.e., the square of imaginary number \[ai\] is \[ - a\] .
Consider the given fraction
 \[ \Rightarrow \dfrac{1}{{{i^3}}}\]
It can be written as
 \[ \Rightarrow \dfrac{1}{{i \cdot i \cdot i}}\]
As we know the value of imaginary number i is \[i = \sqrt { - 1} \] , then
 \[ \Rightarrow \dfrac{1}{{\sqrt { - 1} \cdot \sqrt { - 1} \cdot \sqrt { - 1} }}\]
On multiplying,
 \[ \Rightarrow \dfrac{1}{{{{\left( {\sqrt { - 1} } \right)}^2} \cdot \sqrt { - 1} }}\]
 \[ \Rightarrow \dfrac{1}{{ - 1 \cdot \sqrt { - 1} }}\]
On further simplification we have
 \[ \Rightarrow - \dfrac{1}{{\sqrt { - 1} }}\]
Rationalise the denominator by multiplying both numerator and denominator by \[\sqrt { - 1} \] .
 \[ \Rightarrow - \dfrac{1}{{\sqrt { - 1} }} \times \dfrac{{\sqrt { - 1} }}{{\sqrt { - 1} }}\]
 \[ \Rightarrow - \dfrac{{\sqrt { - 1} }}{{{{\left( {\sqrt { - 1} } \right)}^2}}}\]
 \[ \Rightarrow - \dfrac{{\sqrt { - 1} }}{{ - 1}}\]
Therefore we have
 \[ \Rightarrow \sqrt { - 1} \]
As we know \[\sqrt { - 1} \] is the value of i
 \[\therefore \,\,\dfrac{1}{{{i^3}}} = \sqrt 1 = i\]
Hence, after simplification the value of \[\dfrac{1}{{{i^3}}}\] is \[\sqrt { - 1} \] or \[i\] .
So, the correct answer is “ \[i\] ”.

Note: The complex number is having the real part and imaginary part. The real part is the whole number and the imaginary number is represented by i. and we can also multiply the imaginary number. The imaginary number i is defined as \[\sqrt { - 1} \] . There is a square root for the negative number so we consider it as i.