Simplifying the Exponent Expression

The problem requires simplifying the following mathematical expression involving exponents:

\(\frac{9^{-2} \times 3^{5} \times (5^{-2})^{3}}{ (3^{-3})^{4} \times 5^{4}}\)

To simplify this, we will use the fundamental rules of exponents. Let's break down the process step-by-step.

Understanding Exponent Rules

We need the following exponent rules:

• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Product Rule: \(a^m \times a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Base Conversion: \(9 = 3^2\)

Step-by-Step Simplification

1. Rewrite the base 9: Replace 9 with \(3^2\). The term \(9^{-2}\) becomes \((3^2)^{-2}\).

   Expression: \(\frac{(3^2)^{-2} \times 3^{5} \times (5^{-2})^{3}}{ (3^{-3})^{4} \times 5^{4}}\)

2. Apply the Power of a Power rule: Simplify terms with nested exponents.

   • \((3^2)^{-2} = 3^{2 \times (-2)} = 3^{-4}\)
   • \((5^{-2})^{3} = 5^{-2 \times 3} = 5^{-6}\)
   • \((3^{-3})^{4} = 3^{-3 \times 4} = 3^{-12}\)

   Substitute these back into the expression:

   \(\frac{3^{-4} \times 3^{5} \times 5^{-6}}{ 3^{-12} \times 5^{4}}\)

3. Apply the Product Rule in the numerator: Combine terms with the same base (base 3).

   \(3^{-4} \times 3^{5} = 3^{-4 + 5} = 3^{1}\)

   The expression now is:

   \(\frac{3^{1} \times 5^{-6}}{ 3^{-12} \times 5^{4}}\)

4. Apply the Quotient Rule: Separate terms with base 3 and base 5 and simplify using the quotient rule.

   • For base 3: \(\frac{3^{1}}{3^{-12}} = 3^{1 - (-12)} = 3^{1 + 12} = 3^{13}\)
   • For base 5: \(\frac{5^{-6}}{5^{4}} = 5^{-6 - 4} = 5^{-10}\)

5. Combine the results: Multiply the simplified terms for base 3 and base 5.

   \(3^{13} \times 5^{-10}\)

6. Final Simplified Form: Express the result with positive exponents where possible. \(5^{-10}\) can be written as \(\frac{1}{5^{10}}\).

   The final simplified expression is \(\frac{3^{13}}{5^{10}}\).

Comparing with Options

The simplified result is \(\frac{3^{13}}{5^{10}}\). This matches the form presented in option 2.