Question:
The value of \( \left( i^{18} + \left(\frac{1}{i}\right)^{25} \right)^3 \) is equal to
The value of \( \left( i^{18} + \left(\frac{1}{i}\right)^{25} \right)^3 \) is equal to
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Always reduce powers of \( i \) using modulo 4 before performing operations.
Updated On: May 1, 2026
- \( \frac{1+i}{2} \)
- \( 2 + 2i \)
- \( \frac{1-i}{2} \)
- \( \sqrt{2} - \sqrt{2}i \)
- \( 2 - 2i \)
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The Correct Option is
Solution and Explanation
Concept:
Powers of \( i \) follow cyclic pattern:
\[
i^4 = 1
\]
Step 1: Simplify \( i^{18} \).
\[ 18 \mod 4 = 2 \Rightarrow i^{18} = i^2 = -1 \]
Step 2: Simplify \( \left(\frac{1}{i}\right)^{25} \).
\[ \frac{1}{i} = -i \] \[ (-i)^{25} = (-1)^{25} i^{25} = - i^{25} \] \[ 25 \mod 4 = 1 \Rightarrow i^{25} = i \Rightarrow = -i \]
Step 3: Add terms.
\[ -1 - i \]
Step 4: Cube expression.
\[ (-1 - i)^3 \] First square: \[ (-1 - i)^2 = 1 + 2i + i^2 = 2i \] Then multiply: \[ (2i)(-1 - i) = -2i -2i^2 = -2i + 2 = 2 - 2i \]
Step 1: Simplify \( i^{18} \).
\[ 18 \mod 4 = 2 \Rightarrow i^{18} = i^2 = -1 \]
Step 2: Simplify \( \left(\frac{1}{i}\right)^{25} \).
\[ \frac{1}{i} = -i \] \[ (-i)^{25} = (-1)^{25} i^{25} = - i^{25} \] \[ 25 \mod 4 = 1 \Rightarrow i^{25} = i \Rightarrow = -i \]
Step 3: Add terms.
\[ -1 - i \]
Step 4: Cube expression.
\[ (-1 - i)^3 \] First square: \[ (-1 - i)^2 = 1 + 2i + i^2 = 2i \] Then multiply: \[ (2i)(-1 - i) = -2i -2i^2 = -2i + 2 = 2 - 2i \]
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