Question

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

Answer 1

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,

& \Rightarrow E={{\left( -1 \right)}^{2}} \\\ & \therefore E=1 \\\ \end{aligned}$$ **Hence, the value of the given expression is 1.** **Note:** One must not consider $i$ as any variable. Remember that $i$ always denotes the imaginary number $$\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 $\left( -1 \right)\times \left( -1 \right)=1$.