Question

How do you simplify ${i^{14}}?$

Answer 1

According to the given question, we have to simplify ${i^{14}}$ .
So, first of all we have to understand about the imaginary number $i$ that is mentioned below.
Imaginary number $i$ : an imaginary number is the part of a complex number which we can write like a real number multiplied by the imaginary unit $i.$ where $i = \sqrt { - 1}$ . The imaginary number, when multiplied by itself, gives a negative value.
For example, consider an imaginary number $3i$ , if multiplied by gives $9{i^2}$ or we can write it as -9. Also, 0 is considered as both real number and imaginary number.
So, first of all we have to break the given expression as ${i^{14}}$ in the even powers of $i.$
Now, we have to put the values of the even powers of $i$ as mentioned below.
Formulas used:
$\Rightarrow {i^2} = - 1 \\\ \Rightarrow {i^4} = + 1 \\\$
And so on.

Complete step by step answer:
Step 1:
First of all we have to rewrite the given expression as ${i^{14}}$ in the even powers of $i.$
$\Rightarrow {i^{14}} \\\ \Rightarrow {\left( {{i^4}} \right)^3} \times {i^2} \\\$

Step 2:
Now, we have to put the values of ${i^2}$ and ${i^4}$ as mentioned in the solution hint.
$\Rightarrow {\left( { + 1} \right)^3} \times \left( { - 1} \right)$

Step 3:
Now, we have to simplify the above expression as obtained in the solution step 2.
$\Rightarrow + 1 \times - 1 \\\ \Rightarrow - 1 \\\$
Hence, the simplified value of the given expression as ${i^{14}}$ is -1.

Note: It is necessary to rewrite the given expression in the even power of $i$ . So we can easily put the values of even power of $i$ as mentioned in the solution hint.
It is necessary to know about the imaginary number that is described in the solution hint.