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?

4^3/7^2
1/7^3 . (-4)-^3
7^2/(-4)^3
(-4)^3/7^2

1 answer

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.

  1. Simplifying \( 15^0 \):
    Since any non-zero number raised to the power of 0 is 1, we can simplify \( 15^0 \) to 1: \[ 15^0 = 1 \]

  2. Rewriting the expression:
    The expression simplifies to: \[ 1 \cdot 7^{-2} / (-4)^{-3} = \frac{7^{-2}}{(-4)^{-3}} \]

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

  4. 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:

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