Write the following as product of primes raised to powers and as simply as possible without negative indices and as a single fraction.

$$200 = 2 \times 100$$ $$100 = 2 \times 50$$ $$50 = 2 \times 25$$ $$25 = 5 \times 5$$

$$200 = 2 \times 2 \times 2 \times 5 \times 5 = 2^3 \times 5^2$$

The final answer is $\boxed{2^3 \times 5^2}$.

b) $2\frac{1}{4}$

$$9 = 3^2$$ $$4 = 2^2$$

$$\frac{9}{4} = \frac{3^2}{2^2} = 3^2 \times 2^{-2}$$

The final answer is $\boxed{3^2 \times 2^{-2}}$.

QUESTION 4. Write the following as simply as possible without using negative indices and as a single fraction.

a) $\frac{(2x^{-3})^3 y^{-2}}{x}$

$$8x^{-10} y^{-2} = \frac{8}{x^{10}y^2}$$

The final answer is $\boxed{\frac{8}{x^{10}y^2}}$.

b) $x(2^{-1}y + y^{-1}x^{-1}) + \frac{(xy)^2}{x^2}$

$$2^{-1} = \frac{1}{2}$$ $$y^{-1} = \frac{1}{y}$$ $$x^{-1} = \frac{1}{x}$$

So, the expression inside the parenthesis becomes $\frac{1}{2}y + \frac{1}{y}\frac{1}{x} = \frac{y}{2} + \frac{1}{xy}$.

$$\frac{xy^2 + 2 + 2y^3}{2y}$$

The final answer is $\boxed{\frac{xy^2 + 2y^3 + 2}{2y}}$.