Write the real and imaginary parts of the following
(i) `sqrt(37)+sqrt(-19)`
(ii) `(sqrt(17))/(2)+(2)/(sqrt(70))i`.

Answer

To solve the problem, we will break it down into two parts as given in the question.

### Part (i): Find the real and imaginary parts of \( \sqrt{37} + \sqrt{-19} \)

1. **Identify the terms**: We have \( \sqrt{37} \) which is a real number and \( \sqrt{-19} \) which is an imaginary number.

2. **Rewrite the imaginary part**: We can express \( \sqrt{-19} \) as \( \sqrt{-1} \cdot \sqrt{19} \). Since \( \sqrt{-1} \) is defined as \( i \) (the imaginary unit), we can write:
\[ \sqrt{-19} = i \sqrt{19} \]

3. **Combine the terms**: Now we can write the complex number as:
\[ \sqrt{37} + i \sqrt{19} \]

4. **Identify the real and imaginary parts**: In the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part:
- Real part \( A = \sqrt{37} \)
- Imaginary part \( B = \sqrt{19} \)

### Conclusion for Part (i):
- Real part: \( \sqrt{37} \)
- Imaginary part: \( \sqrt{19} \)

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### Part (ii): Find the real and imaginary parts of \( \frac{\sqrt{17}}{2} + \frac{2}{\sqrt{70}} i \)

1. **Identify the terms**: We have \( \frac{\sqrt{17}}{2} \) which is a real number and \( \frac{2}{\sqrt{70}} i \) which is an imaginary number.

2. **Rewrite the complex number**: The expression is already in the form \( a + bi \):
\[ \frac{\sqrt{17}}{2} + \frac{2}{\sqrt{70}} i \]

3. **Identify the real and imaginary parts**: In the form \( a + bi \):
- Real part \( A = \frac{\sqrt{17}}{2} \)
- Imaginary part \( B = \frac{2}{\sqrt{70}} \)

### Conclusion for Part (ii):
- Real part: \( \frac{\sqrt{17}}{2} \)
- Imaginary part: \( \frac{2}{\sqrt{70}} \)

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