To simplify the expression \( \frac{15^0 \cdot 7^{-2}}{(-4)^{-3}} \) using the Properties of Integer Exponents and convert it to an equivalent expression with only positive exponents, we will follow these steps:
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Evaluate \(15^0\): According to the property of exponents, any non-zero number raised to the power of 0 is 1. \[ 15^0 = 1 \] Therefore, the expression simplifies to: \[ \frac{1 \cdot 7^{-2}}{(-4)^{-3}} = \frac{7^{-2}}{(-4)^{-3}} \]
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Use the property of negative exponents: The rule states that \( x^{-n} = \frac{1}{x^n} \). Thus, we can rewrite the expression as: \[ \frac{7^{-2}}{(-4)^{-3}} = 7^{-2} \cdot (-4)^3 \] We can also rewrite \( 7^{-2} \) as \( \frac{1}{7^2} \): \[ \frac{1}{7^2} \cdot (-4)^3 \]
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Calculate \( (-4)^3 \): \[ (-4)^3 = -64 \]
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Combine these results: \[ \frac{-64}{7^2} = -\frac{64}{49} \]
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Convert to positive exponents: The expression \(-\frac{64}{49}\) can be rewritten with positive exponents: \[ -\frac{64}{7^2} \]
Since the problem does not specify how to treat the negative part, if we just focus on the positive exponent aspect, we can rewrite the final expression as follows while keeping the numerical clarity: \[ -\frac{64}{49}, \text{ or equivalently, } -64 \cdot 7^{-2} \]
Thus, the equivalent expression with only positive exponents is:
\[ \frac{-64}{49} \]
or, in terms of the operation on \( -4 \):
\[ \frac{(-4)^3}{7^2} \]
Please note that depending on how the question interprets "only positive exponents", the expression can retain the negative as shown.