Question

Use the Quotient Property and Negative Exponent Property to generate an equivalent expression for 49√/7^−3
. What will the rational exponent be for the new expression?(1 point)
7^5
Incorrect

Use care when applying the rules of each property, and watch the signs on the exponents.

Answer 1

To simplify the expression \( \frac{49}{7^{-3}} \) using the Quotient Property and the Negative Exponent Property, we can proceed as follows:

1. **Negative Exponent Property** states that \( a^{-n} = \frac{1}{a^n} \). This means:
\[
7^{-3} = \frac{1}{7^3}
\]
Therefore, the original expression can be rewritten as:
\[
\frac{49}{7^{-3}} = 49 \cdot 7^{3}
\]

2. **Expressing 49 as a power of 7**: We know that \( 49 = 7^2 \), thus:
\[
49 \cdot 7^3 = 7^2 \cdot 7^3
\]

3. **Using the Product Property** of exponents: When multiplying powers with the same base, you can add the exponents:
\[
7^2 \cdot 7^3 = 7^{2 + 3} = 7^5
\]

Thus, we find:
\[
\frac{49}{7^{-3}} = 7^5
\]

Now, to answer your question about the rational exponent for the new expression, the exponent is \( 5 \).

So, the equivalent expression is \( 7^5 \) and the rational exponent is \( 5 \).