Question

Evaluate the value of ${{\left( -\sqrt{-1} \right)}^{4n+3}}$ if $n\in N$.

Answer 1

Hint: Just solve the question using normal algebra of exponents and small imaginary concepts like ${{i}^{2}}=-1,{{i}^{4}}=1$. Substitute the imaginary number into the equation and then solve the complex equation.

Complete step-by-step solution -
Definition of i, can be written as:
The solution of the equation: ${{x}^{2}}+1=0$ is i. i is an imaginary number. Any number which has an imaginary number in its representation is called a complex number.
Definition of a complex equation, can be written as: An equation containing complex numbers in it, is called a complex equation.
It is possible to have a real root for complex equations.
Example: $\left( 1+i \right)x+\left( 1+i \right)=0,x= -1$ is the root of the equation.
Given expression of n in the question is in the form:
${{\left( -\sqrt{-1} \right)}^{4n+3}}$
Substitute the imaginary number i into the expression inside the bracket as the imaginary number i is a solution of second degree equation of $x$ :
${{x}^{2}}+1=0\Rightarrow i=\sqrt{-1}$
By substituting this our given expression takes the form of:
${{\left( -i \right)}^{4n+3}}$
Now use a basic concept of exponents in algebra. The identity:
${{a}^{b+c}}={{a}^{b}}.{{a}^{c}}$
By using the above identity our expression takes form of:
${{\left( -i \right)}^{4n+3}}={{\left( -i \right)}^{4n}}.{{\left( -i \right)}^{3}}$
Now using the basic identity of imaginary numbers. The identity:
${{i}^{4}}=1$
By using the above identity, we get
$={{\left( -1 \right)}^{4n}}.{{\left( i \right)}^{4n}}{{\left( -1 \right)}^{3}}{{\left( i \right)}^{3}}$
By using the basic identity of exponents here. The identity:
${{a}^{bc}}={{\left( {{a}^{b}} \right)}^{c}}$
$={{\left( -1 \right)}^{4n}}{{\left( 1 \right)}^{n}}\left( -1 \right){{\left( i \right)}^{3}}$
Now using basic identity if imaginary numbers. The identity:
$\begin{align}
  & {{i}^{3}}=-i \\
 & =1.1.\left( -1 \right).\left( -i \right) \\
\end{align}$
By simplifying the above, we get:
${{\left( -\sqrt{-1} \right)}^{4n+3}}=i$
Then ‘i’ is the value of the required expression.

Note: Be careful while separating terms as if you forgot a minus sign you get -i as answer which is wrong. Here we need to remember the general form of iota and its peoperty.