Apply the Properties of Integer Exponents to generate equivalent expressions to \( \frac{\left(4^{3} \cdot 7^{-2}\right)^{-5}}{7^{9} \cdot 4^{-13}} \) with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents. (1 point) \( \frac{\left(4^{3} \cdot 7^{-2}\right)^{-5}}{7^{9} \cdot 4^{-13}}= \)

To simplify the expression \( \frac{\left(4^{3} \cdot 7^{-2}\right)^{-5}}{7^{9} \cdot 4^{-13}} \) using the properties of integer exponents, we will follow these steps: 1. **Apply the negative exponent rule**: \( a^{-n} = \frac{1}{a^n} \). 2. **Distribute the exponent in the numerator**. 3. **Combine the terms in the numerator and denominator**. 4. **Simplify the expression**. Let's start with the expression: \[ \frac{\left(4^{3} \cdot 7^{-2}\right)^{-5}}{7^{9} \cdot 4^{-13}} \] ### Step 1: Apply the negative exponent rule The numerator can be rewritten as: \[ \left(4^{3} \cdot 7^{-2}\right)^{-5} = \frac{1}{\left(4^{3} \cdot 7^{-2}\right)^{5}} = \frac{1}{4^{15} \cdot 7^{-10}} = \frac{7^{10}}{4^{15}} \] Now, substituting this back into the expression gives: \[ \frac{7^{10}}{4^{15}} \div (7^{9} \cdot 4^{-13}) \] ### Step 2: Rewrite the division as multiplication This can be rewritten as: \[ \frac{7^{10}}{4^{15}} \cdot \frac{1}{7^{9} \cdot 4^{-13}} = \frac{7^{10}}{4^{15}} \cdot \frac{1}{7^{9}} \cdot \frac{1}{4^{-13}} = \frac{7^{10}}{4^{15}} \cdot \frac{4^{13}}{7^{9}} \] ### Step 3: Combine the terms Now we can combine the fractions: \[ \frac{7^{10} \cdot 4^{13}}{4^{15} \cdot 7^{9}} = \frac{4^{13}}{4^{15}} \cdot \frac{7^{10}}{7^{9}} \] ### Step 4: Apply the properties of exponents Using the property \( \frac{a^m}{a^n} = a^{m-n} \): \[ \frac{4^{13}}{4^{15}} = 4^{13-15} = 4^{-2} = \frac{1}{4^{2}} = \frac{1}{16} \] \[ \frac{7^{10}}{7^{9}} = 7^{10-9} = 7^{1} = 7 \] ### Step 5: Combine the results Now we can combine these results: \[ \frac{7}{16} \] Thus, the simplified expression is: \[ \frac{\left(4^{3} \cdot 7^{-2}\right)^{-5}}{7^{9} \cdot 4^{-13}} = \frac{7}{16} \] So the final answer is: \[ \boxed{\frac{7}{16}} \]