Which of the following is an equivalent expression to 15^0 ⋅ 7^−2 / (−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)

Question

Responses

4372
Start Fraction 4 cubed over 7 squared end fraction

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

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

172 ⋅ (−4)−3

Answer 1

To simplify the expression \( 15^0 \cdot 7^{-2} / (-4)^{-3} \) and express it with only positive exponents, we will apply the Properties of Integer Exponents step by step.

1. We know that \( a^0 = 1 \) for any non-zero \( a \). Thus, \( 15^0 = 1 \).

   Therefore, our expression becomes: \[ 1 \cdot 7^{-2} / (-4)^{-3} = 7^{-2} / (-4)^{-3} \]

2. Next, we can rewrite the division of negative exponents. Recall that \( \frac{a^{-m}}{b^{-n}} = \frac{b^n}{a^m} \): \[ 7^{-2} / (-4)^{-3} = 7^{-2} \cdot (-4)^{3} \]

3. We can convert \( 7^{-2} \) to a positive exponent by moving it to the denominator: \[ = \frac{(-4)^3}{7^2} \]

Now, substituting this back, we get that the equivalent expression with only positive exponents is: \[ \frac{(-4)^3}{7^2} \]

This corresponds to the response: \[ \text{Start Fraction } (−4)^3 \text{ over } 7^2 \text{ end fraction} \]

Therefore, the correct response is: \[ \text{Start Fraction } (−4)^3 \text{ over } 7^2 \text{ end fraction} \]