Let's break down and solve the two given expressions, simplifying them into the form $a + bi$ where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

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

1. First, recognize that $\sqrt{-49}$ can be rewritten as $\sqrt{49} \times i = 7i$ because $\sqrt{-1} = i$.

2. Now, simplify the expression: $4 + 7i$ This is already in the form $a + bi$, where $a = 4$ and $b = 7$.

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

1. Distribute the division over both terms in the numerator: $\frac{3}{7} - \frac{8i}{7}$

2. This expression is also already in the form $a + bi$, where $a = \frac{3}{7}$ and $b = -\frac{8}{7}$.

Final Answers:

• a: $4 + 7i$
• b: $\frac{3}{7} - \frac{8}{7}i$

Would you like any further explanation on these steps?

Here are 5 related questions for practice:

1. How do you simplify $\sqrt{-16}$ into an imaginary number?
2. What is the complex form of $6 + \sqrt{-9}$?
3. Can you simplify $\frac{5 - 3i}{2}$?
4. How do you add two complex numbers $(3 + 4i) + (5 - 2i)$?
5. What happens when you multiply $i \times i$?

Tip: When dealing with square roots of negative numbers, always factor out the imaginary unit $i$ from the square root of the positive part.