Question

How do I find the value of a given power of i?

Answer 1

Hint: We have to find the value of a given power of i. We know that the value of i is $\sqrt{-1}$ . i is defined as an imaginary or imaginary unit. The imaginary unit number is used to express the complex numbers. We can find the value of a power of i by taking its square.

Complete step-by-step solution:
We have to find the value of a given power of i. We know that the value of i is $\sqrt{-1}$ . i is defined as an imaginary or imaginary unit. The imaginary unit number is used to express the complex numbers.
We can find the value of a power of i by taking its square.
$\Rightarrow {{i}^{2}}={{\left( \sqrt{-1} \right)}^{2}}=-1$
Let us understand more through the following examples.
First let us see the value of the third power of i.
$\Rightarrow {{i}^{3}}=i\times i\times i={{\left( i \right)}^{2}}\times i=-1\times i=-i$
Now, let us see the value of the fourth power of i.
$\Rightarrow {{i}^{4}}=i\times i\times i\times i={{\left( i \right)}^{2}}\times {{\left( i \right)}^{2}}=-1\times -1=1$
Let us see the value of the fifth power of i.
\[\Rightarrow {{i}^{5}}=i\times i\times i\times i\times i={{\left( i \right)}^{4}}\times i=1\times i=i\]
The sixth power of i can be found as:
\[\Rightarrow {{i}^{6}}=i\times i\times i\times i\times i\times i={{\left( i \right)}^{5}}\times i=i\times i={{i}^{2}}=-1\]
In this way, we can find the value of a given power of i.

Note: Students have a chance of making a mistake by misunderstanding the value of i as $-\sqrt{1}$ . We can see that the odd powers of I will have the value i along with positive or negative sign while for even powers of i, the value will be an integer.