et z_1 =(-i)/(-1+i),z_2=1+i/(1-i) and z_3=1/10[2(i-1)i+(-i+sqrt3)^3]+(1-i)bar((1-i))]

express /bar(z_1z_2)/z_3,z_1barz_2/z3 and barz_1/barz_3z_2 in both polar and standard forms

Let's start by calculating the values of \( z_1 \), \( z_2 \), and \( z_3 \), and then proceed to find the required expressions in both polar and standard forms.

### Step 1: Calculate \( z_1 \), \( z_2 \), and \( z_3 \)

**For \( z_1 = \frac{-i}{-1 + i} \):**

Simplify \( z_1 \) by multiplying the numerator and denominator by the conjugate of the denominator:

\[
z_1 = \frac{-i}{-1 + i} \cdot \frac{-1 - i}{-1 - i} = \frac{i(-1 - i)}{(-1)^2 - (i)^2} = \frac{-i - i^2}{1 + 1} = \frac{-i + 1}{2} = \frac{1 - i}{2}
\]

**For \( z_2 = \frac{1 + i}{1 - i} \):**

Simplify \( z_2 \) similarly by multiplying the numerator and denominator by the conjugate of the denominator:

\[
z_2 = \frac{1 + i}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{(1 + i)^2}{1^2 - i^2} = \frac{1 + 2i + i^2}{1 + 1} = \frac{1 + 2i - 1}{2} = i
\]

**For \( z_3 = \frac{1}{10} \left[ 2(i - 1)i + (-i + \sqrt{3})^3 \right] + (1 - i)\overline{(1 - i)} \):**

Simplify step by step:

1. **Calculate \( (1 - i)\overline{(1 - i)} \):**

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

2. **Calculate \( 2(i - 1)i \):**

\[
2(i - 1)i = 2(i^2 - i) = 2(-1 - i) = -2 - 2i
\]

3. **Calculate \( (-i + \sqrt{3})^3 \):**

First expand \( (-i + \sqrt{3})^3 \):

\[
(-

Express /bar(z_1z_2)/z_3,z_1barz_2/z3 and barz_1/barz_3z_2 in both polar and standard forms, given that z_1 = (-i)/(-1+i), z_2 = 1+i/(1-i) and z_3 = 1/10[2(i-1)i + (-i+sqrt3)^3] + (1-i)bar((1-i))

Explore the detailed solution for expressing complex number expressions in both polar and standard forms. The problem involves manipulating complex numbers using their conjugates, polar representation, and standard algebraic techniques. Suitable for advanced high school or undergraduate students.

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