To solve the given expression, let's first extract the expression from the image:

$\text{Simplify: } \frac{18x^2z^6}{3x^6z^2 \times 2x^5z^7}$

Step-by-Step Simplification

1. Combine the terms in the denominator:

$3x^6z^2 \times 2x^5z^7 = (3 \times 2) (x^6 \times x^5) (z^2 \times z^7) = 6x^{11}z^9$

So, the expression now is:

$\frac{18x^2z^6}{6x^{11}z^9}$

2. Simplify the coefficients (numerical part):

$\frac{18}{6} = 3$

3. Simplify the exponents using the quotient rule:

• For $x$: $x^2 / x^{11} = x^{2 - 11} = x^{-9}$
• For $z$: $z^6 / z^9 = z^{6 - 9} = z^{-3}$

4. Combine everything:

$3 \cdot x^{-9} \cdot z^{-3}$

5. Express with positive exponents:

$\frac{3}{x^9z^3}$

Final Answer:

$\frac{3}{x^9z^3}$

Would you like more details or have any questions about this solution?

Related Questions:

1. What is the quotient rule for exponents and how does it work?
2. How do we handle negative exponents in algebraic expressions?
3. What are the rules for multiplying powers with the same base?
4. How can expressions be simplified when both multiplication and division are involved?
5. What other methods can be used to simplify complex fractions?

Tip:

Always keep track of both the coefficients and the variables when simplifying expressions.