Let's go through the questions one by one and solve them.

Problem 6:

We need to find a number $x$ such that:

$\frac{\left( -\frac{2}{3} \right)^{-3}}{x} = \left( \frac{9}{8} \right)^{-2}$

Steps:

1. Simplify both sides. $\left( -\frac{2}{3} \right)^{-3} = \left( -\frac{3}{2} \right)^3 = -\frac{27}{8}$ And: $\left( \frac{9}{8} \right)^{-2} = \left( \frac{8}{9} \right)^2 = \frac{64}{81}$

2. Now substitute into the equation: $\frac{-\frac{27}{8}}{x} = \frac{64}{81}$

3. Solve for $x$: $x = \frac{-\frac{27}{8}}{\frac{64}{81}} = -\frac{27}{8} \times \frac{81}{64} = -\frac{2187}{512}$

Thus, the number $x$ is $-\frac{2187}{512}$.

Problem 7:

We are asked to find the product of 144 multiplied by $2^{-3}$, $12^{-1}$, and $3^{-2}$.

Steps:

1. Start with 144.
2. Multiply by $2^{-3}$, which is $\frac{1}{2^3} = \frac{1}{8}$.
3. Multiply by $12^{-1}$, which is $\frac{1}{12}$.
4. Multiply by $3^{-2}$, which is $\frac{1}{3^2} = \frac{1}{9}$.

Now compute: $144 \times \frac{1}{8} \times \frac{1}{12} \times \frac{1}{9} = \frac{144}{8 \times 12 \times 9} = \frac{144}{864} = \frac{1}{6}$

Thus, the final product is $\frac{1}{6}$.

Problem 8:

We need to find the number $y$ such that:

$\frac{\left( \frac{3}{4} \right)^{-4} \times \left( -\frac{2}{3} \right)^2}{y} = \left( \frac{3}{5} \right)^{-2}$

Steps:

1. Simplify the expressions. $\left( \frac{3}{4} \right)^{-4} = \left( \frac{4}{3} \right)^4 = \frac{256}{81}$ $\left( -\frac{2}{3} \right)^2 = \frac{4}{9}$ So the product is: $\frac{256}{81} \times \frac{4}{9} = \frac{1024}{729}$ On the right-hand side: $\left( \frac{3}{5} \right)^{-2} = \left( \frac{5}{3} \right)^2 = \frac{25}{9}$

2. Substitute into the equation: $\frac{\frac{1024}{729}}{y} = \frac{25}{9}$

3. Solve for $y$: $y = \frac{\frac{1024}{729}}{\frac{25}{9}} = \frac{1024}{729} \times \frac{9}{25} = \frac{9216}{18225}$

Thus, the number $y$ is $\frac{9216}{18225}$.

Let me know if you need further details or have any questions! Here are some related questions:

1. What is the general rule for negative exponents?
2. How do you simplify fractions raised to a negative power?
3. How do you handle division of fractions?
4. Can you explain the reciprocal relationship in fraction exponents?
5. How does raising a negative fraction to an even power affect the result?

Tip: When working with exponents and fractions, simplifying both the base and exponent first can make the calculations easier!