Question

Solve the following: ${i^{57}} + \dfrac{1}{{{i^{125}}}} = ?$
A) 0
B) $2i$
C) $- 2i$
D) 2

Answer 1

Here we will find the value of the equation given by using the property of iota which is represented by $i$. First, we will use the property that ${i^4} = 1$ so for that, we will make the iota term in the power of $4n$. Then we will solve the iota term by using the exponent property where the exponent with the same base when multiplied there power are added. Finally, we will solve the equation to get the required answer.

Complete step by step solution:
The equation we have to solve is: ${i^{57}} + \dfrac{1}{{{i^{125}}}}$
As we know
$\begin{array}{l}i = \sqrt 1 \\\\{i^2} = - 1\\\\{i^3} = - 1\\\\{i^4} = 1\end{array}$
So we can say,
$\begin{array}{l}{i^{4n + 1}} = i\\\\{i^{4n + 2}} = - 1\\\\{i^{4n + 3}} = - i\\\\{i^{4n + 4}} = 1 = {i^{4n}}\end{array}$
Now we will split the term of iota by using exponent property. Therefore, we get
${i^{57}} + \dfrac{1}{{{i^{125}}}} = {i^{4 \times 14}} \cdot {i^1} + \dfrac{1}{{{i^{4 \times 31}} \cdot {i^1}}}$
Using ${i^{4n}} = 1$ in the above equation, we get
$\Rightarrow {i^{57}} + \dfrac{1}{{{i^{125}}}} = i + \dfrac{1}{i}$
We will multiply and divide the above equation by $i$ in the fraction term. So, we get
$\begin{array}{l} \Rightarrow {i^{57}} + \dfrac{1}{{{i^{125}}}} = i + \dfrac{1}{i} \times \dfrac{i}{i}\\\ \Rightarrow {i^{57}} + \dfrac{1}{{{i^{125}}}} = i + \dfrac{i}{{{i^2}}}\end{array}$
As ${i^2} = - 1$ we can write above term as,
$\Rightarrow {i^{57}} + \dfrac{1}{{{i^{125}}}} = i + \left( { - i} \right)$
Adding and subtracting the terms, we get
$\Rightarrow {i^{57}} + \dfrac{1}{{{i^{125}}}} = 0$

Hence, option (A) is correct.

Note:
The numbers which don’t fall anywhere in the number line are known as imaginary numbers also known as complex numbers. The square root of negative numbers is termed Complex number where we denote them in the form of iota. All the numbers having iota i.e. $i$ in them are known as imaginary numbers. Imaginary numbers is an important concept of mathematics which extends the real number system to the complex number system. We can find the cube root and square root of the complex numbers. Multiplication and division can also be done on complex numbers.