Express the result in the form x+iy, where x,y are real number `i=sqrt(-1)`:
(i) `(-5+3i)(8-7i)`
(ii) `(-sqrt(3)+sqrt(-2))(2sqrt(3)-i)`
(iii) `(sqrt(2)-sqrt(3)i)^(2)`

Answer

Let's solve each part step by step and express the results in the form \( x + iy \), where \( x \) and \( y \) are real numbers.

### (i) \( (-5 + 3i)(8 - 7i) \)

**Step 1:** Use the distributive property (FOIL method) to multiply the two complex numbers.

\[ (-5)(8) + (-5)(-7i) + (3i)(8) + (3i)(-7i) \]

**Step 2:** Calculate each term.

\[ -40 + 35i + 24i - 21i^2 \]

**Step 3:** Recall that \( i^2 = -1 \), so replace \( -21i^2 \) with \( 21 \).

\[ -40 + 35i + 24i + 21 \]

**Step 4:** Combine the real parts and the imaginary parts.

\[ (-40 + 21) + (35i + 24i) = -19 + 59i \]

**Final Result:**

\[ -19 + 59i \]

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### (ii) \( (-\sqrt{3} + \sqrt{-2})(2\sqrt{3} - i) \)

**Step 1:** Rewrite \( \sqrt{-2} \) as \( i\sqrt{2} \).

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

**Step 2:** Use the distributive property (FOIL method).

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

**Step 3:** Calculate each term.

\[ -6 + \sqrt{3}i + 2\sqrt{6}i - 2 \]

**Step 4:** Combine the real parts and the imaginary parts.

\[ (-6 - 2) + (\sqrt{3}i + 2\sqrt{6}i) = -8 + (\sqrt{3} + 2\sqrt{6})i \]

**Final Result:**

\[ -8 + (\sqrt{3} + 2\sqrt{6})i \]

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### (iii) \( (\sqrt{2} - \sqrt{3}i)^2 \)

**Step 1:** Use the formula for squaring a binomial, \( (a - b)^2 = a^2 - 2ab + b^2 \).

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

**Step 2:** Calculate each term.

\[ 2 - 2\sqrt{6}i + 3i^2 \]

**Step 3:** Recall that \( i^2 = -1 \), so replace \( 3i^2 \) with \( -3 \).

\[ 2 - 2\sqrt{6}i - 3 \]

**Step 4:** Combine the real parts and the imaginary parts.

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

**Final Result:**

\[ -1 - 2\sqrt{6}i \]

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