Let's solve each expression step by step.

(a) $\sqrt{200} - \sqrt{32}$

We can simplify both square roots:

• $\sqrt{200} = \sqrt{4 \times 50} = 2\sqrt{50} = 2\sqrt{25 \times 2} = 2 \times 5 \times \sqrt{2} = 10\sqrt{2}$
• $\sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}$

Now subtract the two terms:

$10\sqrt{2} - 4\sqrt{2} = 6\sqrt{2}$

Answer: $6\sqrt{2}$

(b) $(3a^3b^3)(4ab^2)^2$

First, simplify the powers of the second term:

$(4ab^2)^2 = 16a^2b^4$

Now multiply the terms:

$(3a^3b^3) \times (16a^2b^4) = 48a^{3+2}b^{3+4} = 48a^5b^7$

Answer: $48a^5b^7$

(c) $\left(\frac{3x^3y^{-3}}{x^{-2}y^{1/2}}\right)^{-2}$

First, simplify inside the parentheses:

$\frac{3x^3y^{-3}}{x^{-2}y^{1/2}} = 3x^{3 - (-2)}y^{-3 - 1/2} = 3x^{5}y^{-7/2}$

Now apply the $-2$ exponent:

$\left(3x^5y^{-7/2}\right)^{-2} = 3^{-2}x^{-10}y^{7}$

$= \frac{1}{9x^{10}y^{-7}} = \frac{y^7}{9x^{10}}$

Answer: $\frac{y^7}{9x^{10}}$

(d) $\frac{x^2 + 3x + 2}{x^2 - x - 2}$

Factor both the numerator and denominator:

• $x^2 + 3x + 2 = (x + 1)(x + 2)$
• $x^2 - x - 2 = (x - 2)(x + 1)$

Now simplify the fraction:

$\frac{(x + 1)(x + 2)}{(x - 2)(x + 1)} = \frac{x + 2}{x - 2}$

Answer: $\frac{x + 2}{x - 2}$

(e) $\frac{\frac{x^2}{x^2 - 4}}{\frac{x + 1}{x + 2}}$

The denominator $x^2 - 4$ can be factored as $(x - 2)(x + 2)$. So we rewrite the expression:

$\frac{\frac{x^2}{(x - 2)(x + 2)}}{\frac{x + 1}{x + 2}} = \frac{x^2}{(x - 2)(x + 2)} \times \frac{x + 2}{x + 1}$

Cancel out the $(x + 2)$:

$\frac{x^2}{(x - 2)} \times \frac{1}{x + 1} = \frac{x^2}{(x - 2)(x + 1)}$

Answer: $\frac{x^2}{(x - 2)(x + 1)}$

(f) $\frac{\frac{y}{x} - \frac{x}{y}}{\frac{1}{x} + \frac{1}{y}}$

First, simplify the numerator:

$\frac{y}{x} - \frac{x}{y} = \frac{y^2 - x^2}{xy}$

Now simplify the denominator:

$\frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy}$

Now divide the two expressions:

$\frac{\frac{y^2 - x^2}{xy}}{\frac{y + x}{xy}} = \frac{y^2 - x^2}{y + x} = \frac{(y - x)(y + x)}{y + x}$

Cancel out the $(y + x)$:

$y - x$

Answer: $y - x$

Do you need further clarification or details on any part? Here are some related questions:

1. How do you simplify square roots in general?
2. What are the properties of exponents used in simplification?
3. How can you factor quadratic expressions efficiently?
4. How do you simplify complex rational expressions?
5. How do you handle negative exponents in fractions?

Tip: Always check for factoring opportunities in both the numerator and the denominator before simplifying rational expressions!