Question

The value of $(1 + i)^5 (1 - i)^7$ is

• A. $-64$
• B. $-64i$
• C. $64i$
• D. $64$

Hint:

Always look for combinations of $(1+i)(1-i) = 2$ or notice that $(1-i)^2 = -2i$ and $(1+i)^2 = 2i$. Breaking high powers into simple squares makes evaluating complex numbers remarkably swift!

Answer 1

The Correct Option is B

Step 1: Group matching powers.
Write $(1+i)^5(1-i)^7 = [(1+i)(1-i)]^5 (1-i)^2$.

Step 2: Use the conjugate product.
$(1+i)(1-i) = 1 - i^2 = 2$, and $(1-i)^2 = 1 - 2i + i^2 = -2i$.

Step 3: Combine.
$2^5 \times (-2i) = 32 \times (-2i) = -64i$.
\[ \boxed{-64i,\ \text{option 2}} \]