(3) If $$ 8 x+y i=i^{-17}+24 $$, where $$ i^{2}=-1 $$, then $$ x-y=-1 $$ $$ \begin{array}{lll} ext { (a) } 2 & ext { (b) } 4 & ext { (c) }-4 \ ext { (d) } 25\end{array} $$

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(3) If $$ 8 x+y i=i^{-17}+24 $$, where $$ i^{2}=-1 $$, then $$ x-y=-1 $$ $$ \begin{array}{lll}	ext { (a) } 2 & 	ext { (b) } 4 & 	ext { (c) }-4 \ 	ext { (d) } 25\end{array} $$

UpStudy AI · Accepted Answer

To solve the equation $$ 8x + yi = i^{-17} + 24 $$ where $$ i^2 = -1 $$, follow these steps:

1. **Simplify $$ i^{-17} $$:**
   - The powers of $$ i $$ cycle every 4:
     $$
     i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1, \quad \text{and so on}.
     $$
   - To find $$ i^{-17} $$, first find $$ i^{17} $$:
     $$
     17 \div 4 = 4 \text{ with a remainder of } 1 \Rightarrow i^{17} = i^1 = i.
     $$
   - Therefore:
     $$
     i^{-17} = \frac{1}{i^{17}} = \frac{1}{i} = -i.
     $$
     (Since $$ \frac{1}{i} = -i $$)

2. **Rewrite the Original Equation:**
   $$
   8x + yi = -i + 24.
   $$
   
3. **Equate Real and Imaginary Parts:**
   - **Real Part:**
     $$
     8x = 24 \Rightarrow x = 3.
     $$
   - **Imaginary Part:**
     $$
     y = -1.
     $$
   
4. **Compute $$ x - y $$:**
   $$
   x - y = 3 - (-1) = 4.
   $$
   
**Therefore, the correct answer is (b) 4.**

Answer: (b) 4
$$ x - y = 4 $$, so the answer is (b) 4.

(3) If , where , then

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