Question

Evaluate : ${i^{-50}}$

Answer 1

Hint : We have to find the value of ${i^{-50}}$ . We solve this question using the concept of complex numbers . We should also have the knowledge about the values of the powers of iota ( i ) . Firstly we write the term of iota in simplest terms and write its power in terms of $4n{\text{ }} + {\text{ }}a$format . And hence using the values of the power of $i$ , we evaluate the value of ${i^{-50}}$ .

Complete step-by-step answer :
Given :
To evaluate ${i^{-50}}$

The expression can also be written as ,
${i^{-50}} = \dfrac{1}{{[{i^{(50)}}]}}$
Now , we also know that
$i = \sqrt {( - 1)}$
The various values of powers of i are given as :
${i^2} = - 1$
${i^3} = - i$
${i^4} = 1$
${i^5} = i$
${i^6} = - i$
From the values we conclude that the value of power to i repeats after in $\left( {{\text{ }}4n{\text{ }} + {\text{ }}z{\text{ }}} \right)$ terms , where z{\text{ }} = {\text{ }}\left\\{ {{\text{ }}1{\text{ }},{\text{ }}2{\text{ }},{\text{ }}3{\text{ }}} \right\\}
Using these values and concept , we get
${i^{-50}} = \dfrac{1}{{[{i^{(4 \times 12 + 2)}}]}}$
${i^{-50}} = \dfrac{1}{{[{i^{(2)}}]}}$
As , we know ${i^2} = - 1$
${i^{-50}} = - 1$
Hence , we evaluate that the value of ${i^{-50}}$ is $- 1$.
So, the correct answer is “-1”.

Note : A number of the form$a{\text{ }} + {\text{ }}i{\text{ }}b$, where $a$ and $b$ are real numbers , is called a complex number , a is called the real part and b is called the imaginary part of the complex number .
Every real number can be represented in terms of complex numbers but every complex number can’t be represented as a real number . Also every real number is a complex number as every real number can be represented in terms of complex numbers by adding the term of iota with a constant zero multiplied to it .
Since ${b^2} - 4ac$ determines whether the quadratic equation $a{x^2} + bx + c = 0$
If ${b^2} - 4ac < 0$ then the equation has imaginary roots .