To simplify the expression \( 15^0 \cdot 7^{-2} / (-4)^{-3} \) using properties of integer exponents and ensuring all exponents are positive, let's break it down step-by-step.
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Simplifying \( 15^0 \):
Since any non-zero number raised to the power of 0 is 1, we can simplify \( 15^0 \) to 1: \[ 15^0 = 1 \] -
Rewriting the expression:
The expression simplifies to: \[ 1 \cdot 7^{-2} / (-4)^{-3} = \frac{7^{-2}}{(-4)^{-3}} \] -
Using the property of exponents:
We can utilize the property \( a^{-n} = \frac{1}{a^n} \) to rewrite the negative exponents: \[ 7^{-2} = \frac{1}{7^2} \quad \text{and} \quad (-4)^{-3} = \frac{1}{(-4)^3} \] Thus, we can rewrite the expression: \[ \frac{7^{-2}}{(-4)^{-3}} = \frac{\frac{1}{7^2}}{\frac{1}{(-4)^3}} = \frac{(-4)^3}{7^2} \] -
Final expression:
The final expression, with only positive exponents, is: \[ \frac{(-4)^3}{7^2} \]
Thus, the equivalent expression to \( 15^0 \cdot 7^{-2} / (-4)^{-3} \) with only positive exponents is:
\[ (-4)^3 / 7^2 \]
Now, looking through the options provided, the correct answer is:
\(\boxed{\frac{(-4)^3}{7^2}}\)