Question 1209625: Express
\cfrac{1}{1 - \cfrac{1}{2 + \cfrac{1}{i}}}
in the form a+bi, where a and b are real numbers.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to simplify the complex fraction:
1. **Simplify the innermost fraction:**
1/i = (1/i) * (i/i) = i/i² = i/(-1) = -i
2. **Substitute back:**
The expression becomes:
1 / (1 - 1 / (2 - i))
3. **Simplify the fraction in the denominator:**
1 / (2 - i) = (1 / (2 - i)) * ((2 + i) / (2 + i)) = (2 + i) / (4 - i²) = (2 + i) / (4 + 1) = (2 + i) / 5 = 2/5 + (1/5)i
4. **Substitute back again:**
1 / (1 - (2/5 + (1/5)i))
5. **Simplify the denominator:**
1 - (2/5 + (1/5)i) = 1 - 2/5 - (1/5)i = 3/5 - (1/5)i
6. **Rewrite the expression:**
1 / (3/5 - (1/5)i)
7. **Multiply by the conjugate of the denominator:**
(1 / (3/5 - (1/5)i)) * ((3/5 + (1/5)i) / (3/5 + (1/5)i)) = (3/5 + (1/5)i) / ((3/5)² - (1/5)²i²) = (3/5 + (1/5)i) / (9/25 + 1/25) = (3/5 + (1/5)i) / (10/25) = (3/5 + (1/5)i) / (2/5)
8. **Simplify:**
(3/5 + (1/5)i) * (5/2) = 3/2 + (1/2)i
Therefore, the expression in the form a + bi is 3/2 + (1/2)i.
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