Express  (1−3–√i2)165  in terms of  a+bi , where  a,b∈R

To express \((1 - \sqrt{3}i)^2\) in the form \(a + bi\), where \(a, b \in \mathbb{R}\), we need to expand and simplify the given expression.

### Step 1: Expand the expression
We start by expanding \((1 - \sqrt{3}i)^2\) using the binomial theorem:

\[
(1 - \sqrt{3}i)^2 = 1^2 - 2(1)(\sqrt{3}i) + (\sqrt{3}i)^2
\]

### Step 2: Simplify each term
1. \(1^2 = 1\)
2. \(-2(1)(\sqrt{3}i) = -2\sqrt{3}i\)
3. \((\sqrt{3}i)^2 = (\sqrt{3})^2 \times i^2 = 3 \times (-1) = -3\)

### Step 3: Combine the results
Now, combining all these terms:

\[
(1 - \sqrt{3}i)^2 = 1 - 2\sqrt{3}i - 3
\]

This simplifies to:

\[
-2 - 2\sqrt{3}i
\]

So, \((1 - \sqrt{3}i)^2 = -2 - 2\sqrt{3}i\).

### Final Answer
The expression \((1 - \sqrt{3}i)^2\) in the form \(a + bi\) is:

\[
\boxed{-2 - 2\sqrt{3}i}
\]

### Do you want details or have any questions?
Here are 5 related questions to expand your understanding:
1. How would the result change if the original expression was \((1 + \sqrt{3}i)^2\)?
2. What is the magnitude of the complex number \(-2 - 2\sqrt{3}i\)?
3. How can we express \((1 - \sqrt{3}i)^3\) in the form \(a + bi\)?
4. What is the polar form of the complex number \(1 - \sqrt{3}i\)?
5. How does the angle (argument) of \(1 - \sqrt{3}i\) relate to the angle of \(-2 - 2\sqrt{3}i\)?

**Tip:** When dealing with complex numbers, always remember to simplify the imaginary unit \(i^2 = -1\) as it often changes the sign of terms.

Learn how to express (1 - √3i)^2 in the form a + bi with a detailed step-by-step solution. This problem involves using the binomial theorem to expand and simplify complex numbers. Suitable for advanced high school students and anyone interested in complex number arithmetic.

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