The image contains an expression that you need to simplify. It looks like this:

$\left( \frac{6x^{-9} y^8 z^4}{8x^{-2} y^{-8} z^{-3}} \right)^{-1}$

Steps to simplify:

1. Apply the exponent of -1: When you apply the exponent of -1 to a fraction, you invert the fraction. So this becomes:

$\frac{8x^{-2} y^{-8} z^{-3}}{6x^{-9} y^8 z^4}$

2. Simplify powers of the same base: Use the properties of exponents, where $a^m / a^n = a^{m-n}$.

• For $x$: $\frac{x^{-2}}{x^{-9}} = x^{-2 + 9} = x^7$
• For $y$: $\frac{y^{-8}}{y^8} = y^{-8 - 8} = y^{-16}$
• For $z$: $\frac{z^{-3}}{z^4} = z^{-3 - 4} = z^{-7}$

3. Combine everything: Now the expression looks like:

$\frac{8 x^7}{6 y^{16} z^7}$

4. Simplify the constants: $\frac{8}{6} = \frac{4}{3}$.

So the final simplified expression is:

$\frac{4 x^7}{3 y^{16} z^7}$

Final Answer:

$\frac{4x^7}{3y^{16}z^7}$

Let me know if you'd like more details or have further questions!

Related questions:

1. What happens if you have negative exponents on both the numerator and denominator?
2. How can you simplify expressions with multiple variables?
3. Can you explain the properties of exponents in more detail?
4. How do you handle zero exponents in equations?
5. What are some other exponent laws useful for simplifying expressions?

Tip:

Always make sure to rewrite negative exponents as fractions to make simplification easier.