Which of the following is an equivalent expression to 15^0 ⋅ 7^−2/(−4)^−3

Question

with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)
Responses

4372
Start Fraction 4 cubed over 7 squared end fraction

(−4)372
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction

72(−4)3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction

172 ⋅ (−4)−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction
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Answer 1

To simplify the expression \( \frac{15^0 \cdot 7^{-2}}{(-4)^{-3}} \) with positive exponents, we can apply the properties of integer exponents.

1. Recognize that any non-zero number raised to the power of zero is 1: \[ 15^0 = 1 \] Therefore, the expression simplifies to: \[ \frac{1 \cdot 7^{-2}}{(-4)^{-3}} = \frac{7^{-2}}{(-4)^{-3}} \]

2. Use the property \( a^{-n} = \frac{1}{a^n} \): \[ 7^{-2} = \frac{1}{7^2} \quad \text{and} \quad (-4)^{-3} = \frac{1}{(-4)^3} \]

3. Substitute these into the expression: \[ \frac{7^{-2}}{(-4)^{-3}} = \frac{\frac{1}{7^2}}{\frac{1}{(-4)^3}} = \frac{(-4)^3}{7^2} \]

At this point, the expression simplifies to: \[ \frac{(-4)^3}{7^2} \]

To ensure the expression contains only positive exponents, we can write the final form: \[ \frac{(4)^3}{7^2} \] since we want only positive bases.

Thus, among the options provided, the equivalent expression is: \[ \frac{4^3}{7^2} \]

The closest matching option from your responses is: Start Fraction 4 cubed over 7 squared end fraction.