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: Understanding the Question:
The question requires us to evaluate a product of complex numbers raised to high integer powers, specifically $(1 + i)^5$ and $(1 - i)^7$.

Step 2: Key Formula or Approach:
Instead of expanding both brackets fully using long binomial expansions, we can combine the bases using exponents rules by matching their power sizes: $$ (a)^n \cdot (b)^{n+m} = (a \cdot b)^n \cdot (b)^m $$ We also utilize the imaginary unit identity property where $i^2 = -1$.

Step 3: Detailed Explanation:
Let's rewrite the expression to group the components with matching powers of 5: $$ (1 + i)^5 (1 - i)^7 = (1 + i)^5 (1 - i)^5 \cdot (1 - i)^2 $$ Combine the first two terms inside a single power bracket: $$ = [(1 + i)(1 - i)]^5 \cdot (1 - i)^2 $$ The expression inside the square bracket is a product of complex conjugates, which follows the difference of squares rule $(1 - i^2)$: $$ (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2 $$ Now substitute this back and expand the remaining separate squared term $(1 - i)^2$: $$ (1 - i)^2 = 1^2 - 2i + i^2 = 1 - 2i - 1 = -2i $$ Assembling the full product expressions together: $$ = (2)^5 \cdot (-2i) $$ $$ = 32 \cdot (-2i) = -64i $$

Step 4: Final Answer:
The simplified value of the expression is $-64i$, which corresponds to option (B).