Question

If \( i = \sqrt{-1} \), \[ \frac{(i+1)^3}{i-1} = \_\_\_\_\_\_\_\_. \]

• A. \( (i + 1) \)
• B. \( -2 \)
• C. \( (i + 1) \)
• D. \( (i - 1) \)

Hint:

When simplifying complex expressions, especially involving \( i = \sqrt{-1} \), remember to use the conjugate of the denominator to simplify the fraction. This technique is essential for resolving expressions with complex numbers.

Answer 1

The Correct Option is A

We are given the expression: \[ \frac{(i+1)^3}{i-1} \] Step 1: Expand \( (i+1)^3 \)
First, expand \( (i+1)^3 \): \[ (i + 1)^3 = i^3 + 3i^2 + 3i + 1 \] Since \( i^2 = -1 \) and \( i^3 = -i \), we get: \[ (i + 1)^3 = -i - 3 + 3i + 1 = 2i - 2 \] Step 2: Simplify the expression
Now substitute this into the original expression: \[ \frac{(i+1)^3}{i-1} = \frac{2i - 2}{i - 1} \] We will now multiply both the numerator and denominator by the conjugate of \( i - 1 \), which is \( i + 1 \): \[ \frac{2i - 2}{i - 1} \times \frac{i + 1}{i + 1} = \frac{(2i - 2)(i + 1)}{(i - 1)(i + 1)} \] The denominator becomes: \[ (i - 1)(i + 1) = i^2 - 1^2 = -1 - 1 = -2 \] Now expand the numerator: \[ (2i - 2)(i + 1) = 2i^2 + 2i - 2i - 2 = -2 + 2i - 2i - 2 = -4 \] Thus the expression becomes: \[ \frac{-4}{-2} = 2 \] So, we obtain the simplified result of \( 2 \).