Let's go through each expression and simplify it in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

Part a: $4 + \sqrt{49}$

• The square root of 49 is 7, so: $4 + \sqrt{49} = 4 + 7 = 11$
• This is a real number, so the result is $11 + 0i$.

Part b: $\frac{3 - 8i}{7}$

• Distribute the division by 7: $\frac{3}{7} - \frac{8i}{7}$
• The real part is $\frac{3}{7}$, and the imaginary part is $-\frac{8}{7}$, so the result is: $\frac{3}{7} - \frac{8}{7}i$

Part c: $\frac{10 + \sqrt{-75}}{20}$

• First, simplify $\sqrt{-75}$: $\sqrt{-75} = \sqrt{-1} \cdot \sqrt{75} = i \cdot \sqrt{25 \cdot 3} = i \cdot 5\sqrt{3} = 5i\sqrt{3}$
• Now substitute into the expression: $\frac{10 + 5i\sqrt{3}}{20}$
• Simplify by dividing both terms by 20: $\frac{10}{20} + \frac{5i\sqrt{3}}{20} = \frac{1}{2} + \frac{i\sqrt{3}}{4}$
• The result is: $\frac{1}{2} + \frac{i\sqrt{3}}{4}$

Summary of results:

1. $11 + 0i$
2. $\frac{3}{7} - \frac{8}{7}i$
3. $\frac{1}{2} + \frac{i\sqrt{3}}{4}$

Let me know if you'd like further details on any of these steps!

Follow-up Questions:

1. What are the properties of complex numbers in polar form?
2. How can we find the modulus of a complex number?
3. Can you explain how to multiply complex numbers?
4. How do we handle complex numbers with higher roots (e.g., cube roots)?
5. How does Euler’s formula relate to complex numbers?

Tip:

When simplifying square roots of negative numbers, always factor out $\sqrt{-1} = i$, which converts the problem into handling imaginary numbers!