Thank you for the clearer image. Let's go through the expression step-by-step again, with the correct terms.

The given expression is:

$\frac{(a^{-3}b^7c^{-8})^2a}{(a^{-4}b)^{-5}c^{-2}} = a^xb^yc^z$

Let's simplify the numerator and denominator separately.

Numerator Simplification:

$(a^{-3}b^7c^{-8})^2a$

Apply the power to each term inside the parenthesis: $= (a^{-3})^2 (b^7)^2 (c^{-8})^2 a$ $= a^{-6} b^{14} c^{-16} a$

Combine the exponents of $a$: $= a^{-6+1} b^{14} c^{-16}$ $= a^{-5} b^{14} c^{-16}$

Denominator Simplification:

$(a^{-4}b)^{-5}c^{-2}$

Apply the power to each term inside the parenthesis: $= (a^{-4})^{-5} (b)^{-5} c^{-2}$ $= a^{20} b^{-5} c^{-2}$

Combining the Simplified Parts:

Now, the simplified expression is: $\frac{a^{-5} b^{14} c^{-16}}{a^{20} b^{-5} c^{-2}}$

Combine the exponents of like bases: $= a^{-5-20} b^{14-(-5)} c^{-16-(-2)}$ $= a^{-25} b^{14+5} c^{-16+2}$ $= a^{-25} b^{19} c^{-14}$

Final Simplification:

Since the expression is given to be equal to $a^xb^yc^z$, we can equate:

$a^xb^yc^z = a^{-25} b^{19} c^{-14}$

Therefore: $x = -25, \quad y = 19, \quad z = -14$

So the simplified expression in terms of $a$, $b$, and $c$ is: $a^{-25}b^{19}c^{-14}$

Do you want more details on any of these steps, or have any questions? Here are some related questions that might help:

1. How to handle negative exponents in algebraic expressions?
2. What are the laws of exponents used in algebra?
3. How to simplify expressions involving multiple variables and exponents?
4. What are common mistakes to avoid when simplifying algebraic expressions?
5. How can we verify the correctness of algebraic simplifications?
6. What are the properties of exponents that are useful in algebra?
7. How to deal with complex fractions in algebraic expressions?
8. Can you provide more examples of simplifying similar expressions?

Tip: When dealing with exponents, always ensure you correctly apply the laws of exponents and handle negative exponents carefully to avoid mistakes.