The expression in the image is:

$\left( \frac{-2a^3b^2c^0}{3a^2b^3c^7} \right)^{-2}$

Step-by-Step Simplification:

1. Simplify the inside of the fraction:

   $\frac{-2a^3b^2c^0}{3a^2b^3c^7}$

   • $c^0 = 1$, so we can remove $c^0$ from the expression:

     $\frac{-2a^3b^2}{3a^2b^3c^7}$

   • For the variable $a$: $\frac{a^3}{a^2} = a^{3-2} = a^1 = a$

   • For the variable $b$: $\frac{b^2}{b^3} = b^{2-3} = b^{-1} = \frac{1}{b}$

   • The term with $c$ remains $c^{-7}$ (since it's in the denominator).

   Now the simplified fraction is: $\frac{-2a}{3bc^7}$

2. Apply the exponent of $-2$:

   The next step is to apply the power of $-2$ to each part of the fraction: $\left( \frac{-2a}{3bc^7} \right)^{-2}$

   When raising a fraction to a negative power, we invert the fraction and apply the positive power:

   $\left( \frac{-2a}{3bc^7} \right)^{-2} = \left( \frac{3bc^7}{-2a} \right)^2$

3. Square both the numerator and denominator:

   • Numerator: $(3bc^7)^2 = 3^2 \cdot b^2 \cdot (c^7)^2 = 9b^2c^{14}$

   • Denominator: $(-2a)^2 = (-2)^2 \cdot a^2 = 4a^2$

   So the expression becomes: $\frac{9b^2c^{14}}{4a^2}$

Final Answer:

$\frac{9b^2c^{14}}{4a^2}$

Would you like a detailed explanation of any part, or do you have any questions?

Here are 5 related questions for further practice:

1. How would the expression change if the power was $-3$ instead of $-2$?
2. What happens if $c$ was raised to $c^2$ instead of $c^0$ in the numerator?
3. What is the result if the expression is raised to a positive power instead of a negative one?
4. How do you handle a negative exponent on a fraction in general?
5. Can you apply the same process if the base contains multiple variables?

Tip: When simplifying expressions with exponents, always simplify the terms inside the parentheses first before applying the exponent outside.