To simplify the expression \(\sqrt{\frac{49}{7^{-3}}}\), we can use the quotient property and the negative exponent property.
Step 1: Rewriting the expression with properties.
The expression can be rewritten as:
\[ \sqrt{\frac{49}{7^{-3}}} = \sqrt{49} \div \sqrt{7^{-3}} \]
Step 2: Simplifying \(\sqrt{49}\).
We know that:
\[ \sqrt{49} = 7 \]
Step 3: Simplifying \(\sqrt{7^{-3}}\).
Using the property of exponents that states \(\sqrt{a^b} = a^{\frac{b}{2}}\), we have:
\[ \sqrt{7^{-3}} = 7^{-\frac{3}{2}} \]
Step 4: Putting it all together.
Now we can represent \(\sqrt{49}\) over \(\sqrt{7^{-3}}\):
\[ \sqrt{\frac{49}{7^{-3}}} = \frac{7}{7^{-\frac{3}{2}}} \]
Step 5: Applying the Quotient Property of Exponents.
Using the quotient property \(a^m \div a^n = a^{m-n}\):
\[ \frac{7}{7^{-\frac{3}{2}}} = 7^{1 - (-\frac{3}{2})} = 7^{1 + \frac{3}{2}} = 7^{\frac{2}{2} + \frac{3}{2}} = 7^{\frac{5}{2}} \]
So, the simplified expression is:
\[ 7^{\frac{5}{2}} \]
Conclusion: The rational exponent for the new expression is \(\frac{5}{2}\).