Express the result in the form x+iy, where x,y are real number `i=sqrt(-1)`:
(i) `(5+sqrt(2)i)/(1-sqrt(2)i)`
(ii) `(2+i)/((1+i)(1-2i))`.

Answer

Let's solve the given problems step by step.

### Part (i): \((5 + \sqrt{2}i)/(1 - \sqrt{2}i)\)

**Step 1: Rationalize the denominator.**
To eliminate the imaginary part in the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(1 + \sqrt{2}i\).

\[ \frac{(5 + \sqrt{2}i)(1 + \sqrt{2}i)}{(1 - \sqrt{2}i)(1 + \sqrt{2}i)} \]

**Step 2: Calculate the denominator.**
Using the difference of squares formula, we have:

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

**Step 3: Calculate the numerator.**
Now, we expand the numerator:

\[ (5 + \sqrt{2}i)(1 + \sqrt{2}i) = 5 \cdot 1 + 5 \cdot \sqrt{2}i + \sqrt{2}i \cdot 1 + \sqrt{2}i \cdot \sqrt{2}i \]
\[ = 5 + 5\sqrt{2}i + \sqrt{2}i + 2i^2 = 5 + 5\sqrt{2}i + \sqrt{2}i - 2 \]
\[ = (5 - 2) + (5\sqrt{2} + \sqrt{2})i = 3 + 6\sqrt{2}i \]

**Step 4: Combine the results.**
Now, we can write the expression as:

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

**Step 5: Simplify.**
Dividing each term by 3 gives:

\[ 1 + 2\sqrt{2}i \]

Thus, the result in the form \(x + iy\) is:

\[ \boxed{1 + 2\sqrt{2}i} \]

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### Part (ii): \(\frac{(2+i)}{(1+i)(1-2i)}\)

**Step 1: Calculate the denominator.**
First, we expand the denominator:

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

**Step 2: Write the expression.**
Now we have:

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

**Step 3: Rationalize the denominator.**
We multiply the numerator and the denominator by the conjugate of the denominator, \(3 + i\):

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

**Step 4: Calculate the denominator.**
Using the difference of squares formula again:

\[ (3-i)(3+i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]

**Step 5: Calculate the numerator.**
Now, we expand the numerator:

\[ (2+i)(3+i) = 2 \cdot 3 + 2 \cdot i + i \cdot 3 + i \cdot i \]
\[ = 6 + 2i + 3i + i^2 = 6 + 5i - 1 = 5 + 5i \]

**Step 6: Combine the results.**
Now we can write the expression as:

\[ \frac{5 + 5i}{10} \]

**Step 7: Simplify.**
Dividing each term by 10 gives:

\[ \frac{1}{2} + \frac{1}{2}i \]

Thus, the result in the form \(x + iy\) is:

\[ \boxed{\frac{1}{2} + \frac{1}{2}i} \]

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