Apply All Exponent Rules Quick Check

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Question
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)
Responses

(−4)^3/7^2
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction

4^3/7^2
Start Fraction 4 cubed over 7 squared end fraction

1/7^2 ⋅ (−4)^−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction

7^2/(−4)^3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction
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Answer 1

To simplify the expression \( \frac{15^0 \cdot 7^{-2}}{(-4)^{-3}} \) to have only positive exponents, we can apply the properties of exponents step by step:

1. Evaluate \( 15^0 \): Any non-zero number raised to the power of 0 is 1. Therefore: \[ 15^0 = 1 \] So the expression simplifies to: \[ \frac{1 \cdot 7^{-2}}{(-4)^{-3}} = \frac{7^{-2}}{(-4)^{-3}} \]

2. Apply the negative exponent rule: Recall that \( a^{-n} = \frac{1}{a^n} \). Thus, we can rewrite the expression: \[ \frac{7^{-2}}{(-4)^{-3}} = 7^{-2} \cdot (-4)^3 \]

3. Convert \( 7^{-2} \) to positive exponent: \[ 7^{-2} = \frac{1}{7^2} \] Substituting this back into the expression gives: \[ \frac{1}{7^2} \cdot (-4)^3 = \frac{(-4)^3}{7^2} \]

4. Evaluate \( (-4)^3 \): Calculate \( (-4) \times (-4) \times (-4) = -64 \). However, since we need to present this with positive exponents, we leave it as \( (-4)^3 \).

So, the fully simplified expression with only positive exponents is: \[ \frac{(-4)^3}{7^2} \]

Looking at the answer choices provided, the option that matches this simplified form is: \[ \frac{(-4)^3}{7^2} \]

Thus, the correct answer is:

\[ \frac{(-4)^3}{7^2} \]