Let's simplify the given expression:

$\left(\frac{5}{6}\right)^{-1} \cdot \left(\frac{3}{5}\right)^{-2}$

Step 1: Simplify each part

1. $\left(\frac{5}{6}\right)^{-1}$:

   • The negative exponent rule states that $a^{-n} = \frac{1}{a^n}$.
   • Therefore, $\left(\frac{5}{6}\right)^{-1} = \frac{6}{5}$.

2. $\left(\frac{3}{5}\right)^{-2}$:

   • Again, applying the negative exponent rule: $\left(\frac{3}{5}\right)^{-2} = \left(\frac{5}{3}\right)^2$.
   • This becomes $\left(\frac{5}{3}\right)^2 = \frac{25}{9}$.

Step 2: Multiply the results

Now, multiply the two results:

$\frac{6}{5} \cdot \frac{25}{9} = \frac{6 \times 25}{5 \times 9} = \frac{150}{45}$

Step 3: Simplify the fraction

$\frac{150}{45} = \frac{150 \div 15}{45 \div 15} = \frac{10}{3}$

Final Answer:

$\left(\frac{5}{6}\right)^{-1} \cdot \left(\frac{3}{5}\right)^{-2} = \frac{10}{3}$

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Would you like more details on any step or have any further questions?

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Here are some related questions:

1. How do negative exponents affect fractions?
2. What is the rule for multiplying fractions with negative exponents?
3. Can you explain how to simplify complex fractions step by step?
4. How does the exponent rule apply to terms with mixed numbers?
5. How would you simplify expressions with multiple negative exponents?

Tip: When dealing with negative exponents, always remember the rule: $a^{-n} = \frac{1}{a^n}$. This will help you transform the negative exponent into a positive one for easier calculation.