Express the following in the form $a + ib$: $(1 - i)^{4}$

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Accepted Answer

> [!TLDR]
> Expand $(1-i)^4$ by first squaring $(1-i)$ twice, using $(a-b)^2 = a^2 - 2ab + b^2$ and remembering that $i^2 = -1$. The result is $\boxed{-4}$.

When we raise a complex number to a power, we're repeatedly multiplying it by itself. The key insight here is that we don't need to multiply $(1-i)$ by itself four times in a row—that would be tedious and error-prone. Instead, we can use the fact that $(1-i)^4 = [(1-i)^2]^2$, squaring twice. Each time we square, we apply the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$ and use the fundamental property $i^2 = -1$ to simplify.

This approach works because exponentiation is just repeated multiplication, and grouping the operations strategically minimizes our work.

## Step-by-step solution

1. **First squaring: compute $(1-i)^2$**

Using the binomial expansion:
   $$
   (1-i)^2 = 1^2 - 2(1)(i) + i^2 = 1 - 2i + i^2
   $$
   
   Since $i^2 = -1$:
   $$
   (1-i)^2 = 1 - 2i - 1 = -2i
   $$

2. **Second squaring: compute $[(1-i)^2]^2 = (-2i)^2$**

Now we square the result from step 1:
   $$
   (-2i)^2 = (-2)^2 \cdot i^2 = 4 \cdot i^2
   $$
   
   Again using $i^2 = -1$: …

Q.Express the following in the form $a + ib$ : $(1 - i)^{4}$

Concept understanding — Complex Number Arithmetic

Complex Number Arithmetic: A First Look

Why Bother?

Arithmetic Operations

Addition and Subtraction

Multiplication

Division

The Big Picture …

Step-by-step solution

Q.Express the following in the form a+iba + iba+ib: (1−i)4(1 - i)^{4}(1−i)4

Step-by-step solution

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Q.Express the following in the form $a + ib$ : $(1 - i)^{4}$

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