Let's simplify and solve the questions step by step:

Question 3:

Simplify the following without a calculator:

a. $\frac{\sqrt{18}}{\sqrt{8}}$

$\sqrt{\frac{18}{8}} = \sqrt{\frac{9}{4}} = \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2}.$

Answer: $\frac{3}{2}$.

b. $\frac{\sqrt{75}}{\sqrt{12}}$

$\sqrt{\frac{75}{12}} = \sqrt{\frac{25}{4}} = \frac{\sqrt{25}}{\sqrt{4}} = \frac{5}{2}.$

Answer: $\frac{5}{2}$.

c. $a^2 b^3 \cdot a^{-5} b^2 = a^{2 - 5} \cdot b^{3 + 2} = a^{-3} \cdot b^5$.

Using positive exponents: $a^{-3} \cdot b^5 = \frac{b^5}{a^3}.$

Answer: $\frac{b^5}{a^3}$.

d. $\frac{15x^4y}{3x^2y^2}$

Simplify coefficients: $\frac{15}{3} = 5.$

Simplify $x^4 / x^2 = x^{4 - 2} = x^2$.

Simplify $y / y^2 = y^{1 - 2} = y^{-1} = \frac{1}{y}$.

$\text{Result: } 5x^2 \cdot \frac{1}{y} = \frac{5x^2}{y}.$

Answer: $\frac{5x^2}{y}$.

Question 4:

If $a = -4$ and $b = -5$, find the values of:

a. $|3a| - |2b|$

Calculate $|3a|$: $3a = 3(-4) = -12, \quad |3a| = 12.$

Calculate $|2b|$: $2b = 2(-5) = -10, \quad |2b| = 10.$

$|3a| - |2b| = 12 - 10 = 2.$

Answer: $2$.

b. $|b - a| - b$

Calculate $b - a$: $b - a = -5 - (-4) = -5 + 4 = -1, \quad |b - a| = |-1| = 1.$

$|b - a| - b = 1 - (-5) = 1 + 5 = 6.$

Answer: $6$.

Would you like further explanations for any part of this?
Here are related questions for practice:

1. Simplify $\frac{\sqrt{50}}{\sqrt{2}}$.
2. Solve $x^3y^{-2} \cdot x^{-1}y^4$ using positive exponents.
3. Evaluate $|4a| - |b|$ when $a = -3$ and $b = 6$.
4. Simplify $\frac{20x^5}{5x^2}$.
5. Rewrite $(a^2b^{-3})^2$ using positive exponents.

Tip: When simplifying radicals, always look for perfect square factors to simplify efficiently.