To simplify the expression \( \frac{15^0 \cdot 7^{-2}}{(-4)^{-3}} \) using the properties of integer exponents and ensure that we have only positive exponents, we can follow these steps:
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Evaluate \( 15^0 \): Any non-zero number raised to the power of 0 is 1. Thus, \( 15^0 = 1 \).
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Substitute \( 15^0 \) into the expression: \[ \frac{1 \cdot 7^{-2}}{(-4)^{-3}} = \frac{7^{-2}}{(-4)^{-3}} \]
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When you divide by \( (-4)^{-3} \), you can apply the property of exponents that states \( a^{-n} = \frac{1}{a^n} \): \[ \frac{7^{-2}}{(-4)^{-3}} = 7^{-2} \cdot (-4)^{3} = \frac{(-4)^{3}}{7^{2}} \]
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Now, we have an expression with only positive exponents: \[ \frac{(-4)^{3}}{7^{2}} \]
From the given responses, we look for an expression that matches our simplified form:
- \( \frac{(-4)^{3}}{7^{2}} \)
The corresponding option from the list is:
- \( \frac{(-4)^{3}}{7^{2}} \)
Therefore, the equivalent expression to \( \frac{15^0 \cdot 7^{-2}}{(-4)^{-3}} \) with only positive exponents is:
\[ \frac{(-4)^{3}}{7^{2}} \]