Question

Use the Quotient Property and Negative Exponent Property to generate an equivalent expression for: the square root of 49/7^−3
. What will the rational exponent be for the new expression?(1 point)
__^7
Use care when applying the rules of each property, and watch the signs on the exponents.

ITS NOT 5 THE ANSWER IS NOT 5 YOU ARE WRONG IF YOU GET THAT ANSWER

Answer 1

To simplify the expression \(\sqrt{\frac{49}{7^{-3}}}\) using the Quotient Property and Negative Exponent Property, let's break it down step by step.

1. **Apply the Quotient Property**:
\[
\sqrt{\frac{49}{7^{-3}}} = \frac{\sqrt{49}}{\sqrt{7^{-3}}}
\]

2. **Calculate \(\sqrt{49}\)**:
\[
\sqrt{49} = 7
\]
So, we have:
\[
\frac{7}{\sqrt{7^{-3}}}
\]

3. **Apply the Negative Exponent Property**:
Recall that \(\sqrt{a^{-n}} = a^{-\frac{n}{2}}\). Therefore:
\[
\sqrt{7^{-3}} = (7^{-3})^{\frac{1}{2}} = 7^{-\frac{3}{2}}
\]

4. **Combine the expression**:
Substitute this back into the expression:
\[
\frac{7}{7^{-\frac{3}{2}}} = 7^{1 - \left(-\frac{3}{2}\right)} = 7^{1 + \frac{3}{2}} = 7^{\frac{2}{2} + \frac{3}{2}} = 7^{\frac{5}{2}}
\]

So, the equivalent expression is \(7^{\frac{5}{2}}\).

To answer the question about the rational exponent for the new expression:
The rational exponent is \(\frac{5}{2}\).

In summary:
\[
\sqrt{\frac{49}{7^{-3}}} = 7^{\frac{5}{2}}
\]
Rational exponent: \(\frac{5}{2}\)