Question

Which property of exponents was used to generate the equivalent expression numerator of 15 to the zero power times 7 to the negative power with a denominator of (negative 4) to the negatives third power with only positive exponents that has been generated by applying the Properties of Integer Exponents?

Answer 1

To rewrite the expression \( \frac{15^0 \cdot 7^{-1}}{(-4)^{-3}} \) using only positive exponents, we need to recall the properties of exponents that apply:

1. **Any number to the zero power is equal to 1:** \( x^0 = 1 \) for any non-zero \( x \).
2. **Negative exponent rule:** \( x^{-n} = \frac{1}{x^n} \) for any non-zero \( x \) and integer \( n \).

Using these properties, we can simplify the expression step by step:

1. Starting with the numerator:
- \( 15^0 = 1 \)
- \( 7^{-1} = \frac{1}{7} \)
- Therefore, \( 15^0 \cdot 7^{-1} = 1 \cdot \frac{1}{7} = \frac{1}{7} \)

2. Now for the denominator:
- \( (-4)^{-3} = \frac{1}{(-4)^3} = \frac{1}{-64} \)

3. This leads us to rewrite the full expression:
\[
\frac{15^0 \cdot 7^{-1}}{(-4)^{-3}} = \frac{\frac{1}{7}}{\frac{1}{-64}} = \frac{1}{7} \cdot (-64) = \frac{-64}{7}
\]

In conclusion, the properties used to rewrite the original expression into one with only positive exponents were the zero exponent property and the negative exponent property.