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?

Answer 1

To simplify the expression \( \frac{15^0 \cdot 7^{-2}}{(-4)^{-3}} \) and rewrite it using only positive exponents, we follow these steps:

1. Simplify \( 15^0 \): \[ 15^0 = 1 \] Thus, the expression becomes: \[ \frac{1 \cdot 7^{-2}}{(-4)^{-3}} = \frac{7^{-2}}{(-4)^{-3}} \]

2. Apply the negative exponent rule: Recall that \( a^{-n} = \frac{1}{a^n} \). Therefore: \[ 7^{-2} = \frac{1}{7^2} \quad \text{and} \quad (-4)^{-3} = \frac{1}{(-4)^3} \] Substituting these into our expression gives: \[ \frac{7^{-2}}{(-4)^{-3}} = \frac{\frac{1}{7^2}}{\frac{1}{(-4)^3}} = \frac{1}{7^2} \cdot \frac{(-4)^3}{1} = \frac{(-4)^3}{7^2} \]

3. Evaluate \((-4)^3\): \[ (-4)^3 = -64 \] This results in: \[ \frac{-64}{7^2} \]

4. Keeping positive exponents: \((7^2)\) remains as is because it has a positive exponent.

Finally, we can write \( 7^2 \) value: \[ 7^2 = 49 \]

Thus the expression in terms of only positive exponents is: \[ -\frac{64}{49} \]

So, the equivalent expression to \( 15^0 \cdot 7^{-2} / (-4)^{-3} \) with only positive exponents is: \[ \frac{-64}{49} \]