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

To solve the problem, we need to first express \( z_1 \), \( z_2 \), and \( z_3 \) in their standard forms (rectangular forms) and then compute the required expressions.

### Step 1: Simplify \( z_1 = \frac{-i}{-1 + i} \)

Simplify the expression by multiplying the numerator and the denominator by the conjugate of the denominator:

\[
z_1 = \frac{-i}{-1 + i} \times \frac{-1 - i}{-1 - i} = \frac{-i(-1 - i)}{(-1 + i)(-1 - i)}
\]

The denominator is:
\[
(-1 + i)(-1 - i) = 1 + 1 = 2
\]

The numerator is:
\[
-i(-1 - i) = i + 1
\]

So,
\[
z_1 = \frac{1 + i}{2} = \frac{1}{2} + \frac{i}{2}
\]

### Step 2: Simplify \( z_2 = \frac{1+i}{1-i} \)

Similarly, simplify by multiplying by the conjugate of the denominator:

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

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

First, simplify each term:

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

For the cubic term, \((-i + \sqrt{3})^3\):

Let \( w = -i + \sqrt{3} \). Then:
\[
w^2 = (-i + \sqrt{3})^2 = (-i)^2 + 2(-i)(\sqrt{3}) + (\sqrt{3})^2 = -1 - 2i\sqrt{3} + 3 = 2 - 2i\sqrt{3}
\]

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

Expanding:
\[
w^3 = 2(-i + \sqrt{3}) - 2i\sqrt{3}(-i + \sqrt{3})
= -2i + 2\sqrt{3} - 6 + 2i^2 = -2i + 2\sqrt{3} - 6 + 2i(-i) + 6\sqrt{3} = (-6 + 2i + 6) + (2i - 6i) = -2\sqrt{3}i
\]

So:
\[
(-i + \sqrt{3})^3 = -6 + 2\sqrt{3} + i(-2 - 6i)
\]

Adding the above with \( -2 - 2i + (-6 + 2\sqrt{3}) + i(\overline{1 - i}) \) we add the following:

Conjugate: \( \overline{(1 - i)} = 1 + i \):

\( = (-6 + 2\sqrt{3}) = \) equals:

### Expression forms:

Express in simplified:

\[
z_3 = -6 + 2\sqrt{3} + (1+i)(1-i)
\]

Combining all of these to a real simplified expression on both \( /bar(z_1z_2)/z_3,z_1barz_2/z3 \).

Tip: Solve math on both formats by steps.

Express |z_1z_2|/z_3, z_1bar(z_2)/z_3, and bar(z_1)/(bar(z_3)z_2) in both polar and standard forms, where 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 how to express complex number operations involving z_1 = (-i)/(-1+i), z_2 = (1+i)/(1-i), and z_3 in both polar and rectangular forms. This detailed step-by-step guide covers key concepts like modulus, conjugation, and the use of De Moivre's Theorem. Suitable for undergraduate students.

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