Question

What is ${{i}^{4}}$?

Answer 1

Hint: Assume the given expression as E. Now, write the given expression as ${{\left( {{i}^{2}} \right)}^{2}}$ by using the formula of exponent \[{{a}^{m\times n}}={{\left( {{a}^{m}} \right)}^{n}}\] where ‘a’ is called the base and m and n are the exponents. Consider $i$ as the imaginary number $\sqrt{-1}$ and simplify the expression to get the answer.

Complete step by step solution:
Here we have been provided with the expression ${{i}^{4}}$ and we have been to find its value. Let us assume the given expression as ‘E’. So we have,
\[\Rightarrow E={{i}^{4}}\]
We can write the above expression as:
\[\Rightarrow E={{i}^{2\times 2}}\]
Applying the formula of exponents given as \[{{a}^{m\times n}}={{\left( {{a}^{m}} \right)}^{n}}\], where ‘a’ is called the base and m and n are the exponents, so we get,
\[\Rightarrow E={{\left( {{i}^{2}} \right)}^{2}}\]
Now, here we can see that in the above expression we have an alphabet $i$, actually it is the notation for the imaginary number \[\sqrt{-1}\]. $i$ is the solution of the quadratic equation . There are no real solutions of this quadratic equation and therefore the concept of imaginary numbers and complex numbers arises. A complex number is written in general form as: - \[z=a+ib\], where ‘z’ is the notation of complex numbers, ‘a’ is the real part and ‘b’ is the imaginary part.
$\because i=\sqrt{-1}$
On squaring both the sides we get,
$\Rightarrow {{i}^{2}}=-1$
Substituting the above value in the expression E we get,
\[\begin{align}
 & \Rightarrow E={{\left( -1 \right)}^{2}} \\
 & \therefore E=1 \\
\end{align}\]

Hence, the value of the given expression is 1.

Note: One must not consider $i$ as any variable. Remember that $i$ always denotes the imaginary number \[\sqrt{-1}\] in the topic ‘complex numbers’. You must remember the formulas of the topic ‘exponents and power’ like: - \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}},{{a}^{m}}\div {{a}^{n}}={{a}^{m-n}},{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], because these formulas are frequently used in other topics of mathematics. You can also write the expression ${{i}^{4}}$ as ${{i}^{2}}\times {{i}^{2}}$ and then substitute its value to get the product $\left( -1 \right)\times \left( -1 \right)=1$.