Question

By simplifying the expression: $$ i^{18} - 3i^7 + i^2(1 + i^4)(i^{22}) $$ we get:

• A. $-1 + 3i$
• B. $1 - 3i$
• C. $1 + 3i$
• D. $-1 - 3i$

Hint:

Remember that \( i^4 = 1 \), so powers of \( i \) repeat every 4 steps. Always reduce large powers modulo 4.

Answer 1

The Correct Option is C

We use the cyclic nature of powers of \( i \): \[ i^1 = i,\quad i^2 = -1,\quad i^3 = -i,\quad i^4 = 1,\quad i^5 = i,\ldots \] So, \[ i^{18} = i^{(4 \times 4 + 2)} = i^2 = -1\\ i^7 = i^3 = -i\\ i^4 = 1 \Rightarrow 1 + i^4 = 1 + 1 = 2\\ i^2 = -1,\quad i^{22} = i^2 = -1 \] Now substitute: \[ i^{18} - 3i^7 + i^2(1 + i^4)(i^{22}) = -1 - 3(-i) + (-1)(2)(-1) \] \[ = -1 + 3i + 2 = 1 + 3i \]