Question

Evaluate ${\left[i^{19}+{\left(\frac{1}{i}\right)}^{25}\right]}^2$

• A. $4$
• B. $-4$
• C. $5$
• D. $-5$

Answer 1

The correct option is B

$-4$

Simplify the given expression

Given:

${\left[i^{19}+{\left(\frac{1}{i}\right)}^{25}\right]}^2$

$\because {\mathit{i}}^{\mathbf{4}}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}{\mathit{i}}^{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{1}$

$\Rightarrow {\left[{\left(i^4\right)}^4.i^3+{\left(\frac{1}{i}\right)}^{25}\right]}^2$

$\Rightarrow {\left[1^4.i^3+\left(-i^{25}\right)\right]}^2$

$\Rightarrow {\left[i^3-i^{24}.i\right]}^2$

$\Rightarrow {\left[i^2.i-{\left(i^4\right)}^6.i\right]}^2$

$\Rightarrow {\left[-i-i\right]}^2$ $\because {\mathit{i}}^{\mathbf{4}}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{}{\mathit{i}}^{\mathbf{2}}\mathbf{=}\mathbf{-}\mathbf{1}$

$\Rightarrow {\left(-2i\right)}^2$

$\Rightarrow 4i^2$

$\Rightarrow \mathbf{-}\mathbf{4}$

Hence, Option ‘B’ is Correct.