Which of the following is an equivalent expression to $$ \frac{15^{0} \cdot 7^{-2}}{(-4)^{-3}} $$ with only positive exponents that has been generated by applying the Properties of Integer Exponents? (1 point) $$ \frac{4^{3}}{7^{2}} $$ $$ \frac{1}{7^{2} \cdot(-4)^{-3}} $$ $$ \frac{7^{2}}{(-4)^{3}} $$ $$ \frac{(-4)^{3}}{7^{2}} $$

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The correct answer is $$ \frac{(-4)^{3}}{7^{2}} $$. To simplify the expression $$ \frac{15^{0} \cdot 7^{-2}}{(-4)^{-3}} $$ and express it with only positive exponents, we can apply the properties of integer exponents. 1. **Evaluate $$ 15^{0} $$**: $$ 15^{0} = 1 $$ So, the expression simplifies to: $$ \frac{1 \cdot 7^{-2}}{(-4)^{-3}} = \frac{7^{-2}}{(-4)^{-3}} $$ 2. **Apply the property of negative exponents**: Recall that $$ a^{-n} = \frac{1}{a^{n}} $$. Therefore: $$ 7^{-2} = \frac{1}{7^{2}} \quad \text{and} \quad (-4)^{-3} = \frac{1}{(-4)^{3}} $$ Thus, we can rewrite the expression: $$ \frac{7^{-2}}{(-4)^{-3}} = \frac{\frac{1}{7^{2}}}{\frac{1}{(-4)^{3}}} $$ 3. **Dividing by a fraction**: Dividing by a fraction is equivalent to multiplying by its reciprocal: $$ \frac{1}{7^{2}} \cdot (-4)^{3} = \frac{(-4)^{3}}{7^{2}} $$ Now we have the expression: $$ \frac{(-4)^{3}}{7^{2}} $$ This expression has only positive exponents. Now, let's compare this with the options provided: - $$ \frac{4^{3}}{7^{2}} $$ - $$ \frac{1}{7^{2} \cdot(-4)^{-3}} $$ - $$ \frac{7^{2}}{(-4)^{3}} $$ - $$ \frac{(-4)^{3}}{7^{2}} $$ The equivalent expression we derived is: $$ \frac{(-4)^{3}}{7^{2}} $$ Thus, the correct answer is: $$ \boxed{\frac{(-4)^{3}}{7^{2}}} $$

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