Which of the following is an equivalent expression to 15^0 ⋅ 7^−2 over (−4)^−3

Question

with only positive exponents that has been generated by applying the Properties of Integer Exponents?

options:

1 over 7^2 ⋅ (−4)^−3

4^3 over 7^2

7^2 over (−4)^3

(−4)^3 over 7^2

Answer 1

To simplify the expression \( \frac{15^0 \cdot 7^{-2}}{(-4)^{-3}} \) using the properties of integer exponents, we proceed as follows:

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

2. Apply the negative exponent rule, which states that \( x^{-n} = \frac{1}{x^n} \):

   • For \( 7^{-2} \): \[ 7^{-2} = \frac{1}{7^2} \]
   • For \( (-4)^{-3} \): \[ (-4)^{-3} = \frac{1}{(-4)^3} \]

3. Rewrite the expression using these rules: \[ \frac{7^{-2}}{(-4)^{-3}} = \frac{\frac{1}{7^2}}{\frac{1}{(-4)^3}} \]

4. When you divide by a fraction, you multiply by its reciprocal: \[ = \frac{1}{7^2} \cdot (-4)^3 = \frac{(-4)^3}{7^2} \]

Now, expressing everything with positive exponents, the equivalent expression is: \[ \frac{(-4)^3}{7^2} \]

From the options provided, the closest equivalent expression that matches our result is:

(−4)^3 over 7^2