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.
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We know that \( a^0 = 1 \) for any non-zero \( a \). Thus, \( 15^0 = 1 \).
Therefore, our expression becomes: \[ 1 \cdot 7^{-2} / (-4)^{-3} = 7^{-2} / (-4)^{-3} \]
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Next, we can rewrite the division of negative exponents. Recall that \( \frac{a^{-m}}{b^{-n}} = \frac{b^n}{a^m} \): \[ 7^{-2} / (-4)^{-3} = 7^{-2} \cdot (-4)^{3} \]
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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: \[ \text{Start Fraction } (−4)^3 \text{ over } 7^2 \text{ end fraction} \]