Write the following in the form x+iy:
(i) `(3+2i)(2-i)`
(ii) `2i^(2)+6i^(3)+3i^(16)-6i^(19)+4i^(25)`.
(iii) `((3-2i)(2+3i))/((1+2i)(2-i))`.

Answer

Let's solve the given problems step by step.

### (i) \((3 + 2i)(2 - i)\)

**Step 1:** Use the distributive property (FOIL method) to expand the expression.

\[ (3 + 2i)(2 - i) = 3 \cdot 2 + 3 \cdot (-i) + 2i \cdot 2 + 2i \cdot (-i) \]

**Step 2:** Calculate each term.

\[ = 6 - 3i + 4i - 2i^2 \]

**Step 3:** Substitute \(i^2 = -1\).

\[ = 6 - 3i + 4i + 2 \]

**Step 4:** Combine like terms.

\[ = (6 + 2) + (-3i + 4i) = 8 + i \]

**Final Answer:** The expression in the form \(x + iy\) is \(8 + i\).

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### (ii) \(2i^2 + 6i^3 + 3i^{16} - 6i^{19} + 4i^{25}\)

**Step 1:** Substitute the powers of \(i\) using the known values:
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\) (and thus \(i^n\) cycles every 4)

Calculate each term:
- \(i^{16} = (i^4)^4 = 1\)
- \(i^{19} = i^{4 \cdot 4 + 3} = -i\)
- \(i^{25} = i^{4 \cdot 6 + 1} = i\)

**Step 2:** Substitute these values into the expression.

\[ = 2(-1) + 6(-i) + 3(1) - 6(-i) + 4(i) \]

**Step 3:** Simplify each term.

\[ = -2 - 6i + 3 + 6i + 4i \]

**Step 4:** Combine like terms.

\[ = (-2 + 3) + (-6i + 6i + 4i) = 1 + 4i \]

**Final Answer:** The expression in the form \(x + iy\) is \(1 + 4i\).

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### (iii) \(\frac{(3 - 2i)(2 + 3i)}{(1 + 2i)(2 - i)}\)

**Step 1:** Calculate the numerator.

\[ (3 - 2i)(2 + 3i) = 3 \cdot 2 + 3 \cdot 3i - 2i \cdot 2 - 2i \cdot 3i \]
\[ = 6 + 9i - 4i - 6i^2 \]
\[ = 6 + 5i + 6 = 12 + 5i \]

**Step 2:** Calculate the denominator.

\[ (1 + 2i)(2 - i) = 1 \cdot 2 + 1 \cdot (-i) + 2i \cdot 2 + 2i \cdot (-i) \]
\[ = 2 - i + 4i - 2i^2 \]
\[ = 2 + 3i + 2 = 4 + 3i \]

**Step 3:** Now we have \(\frac{12 + 5i}{4 + 3i}\). To simplify, multiply the numerator and denominator by the conjugate of the denominator.

\[ \frac{(12 + 5i)(4 - 3i)}{(4 + 3i)(4 - 3i)} \]

**Step 4:** Calculate the new numerator.

\[ = 12 \cdot 4 + 12 \cdot (-3i) + 5i \cdot 4 + 5i \cdot (-3i) \]
\[ = 48 - 36i + 20i - 15 \]
\[ = (48 - 15) + (-36i + 20i) = 33 - 16i \]

**Step 5:** Calculate the new denominator.

\[ = 4^2 - (3i)^2 = 16 + 9 = 25 \]

**Step 6:** Combine the results.

\[ \frac{33 - 16i}{25} = \frac{33}{25} - \frac{16}{25}i \]

**Final Answer:** The expression in the form \(x + iy\) is \(\frac{33}{25} - \frac{16}{25}i\).

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