To generate an equivalent expression for \(\frac{\sqrt{47}}{7^{-3}}\) using the quotient property and the negative exponent property, we can start by rewriting the expression.
Recall that the negative exponent property states that \(a^{-n} = \frac{1}{a^n}\). Therefore, we can rewrite the denominator \(7^{-3}\) as follows:
\[ \frac{\sqrt{47}}{7^{-3}} = \sqrt{47} \cdot 7^{3} \]
Next, we can simplify this further if necessary, but we are interested in expressing it in terms of rational exponents. The square root can be rewritten using rational exponents as follows:
\[ \sqrt{47} = 47^{1/2} \]
Now we can combine these two parts into a single expression:
\[ \sqrt{47} \cdot 7^{3} = 47^{1/2} \cdot 7^{3} \]
We can express this as:
\[ \frac{47^{1/2}}{7^{-3}} \rightarrow 47^{1/2} \cdot 7^{3} \]
Now, since both \(47^{1/2}\) and \(7^{3}\) are in the form of \(a^{m}\) and \(b^{n}\), we do not need to further combine them; however, we can express the entire expression as one single term, if necessary, using multiplication:
\[ = \frac{47^{1/2} \cdot 7^{3}}{1} \equiv (47^{1/2} \cdot 7^{3}) \]
In rational exponent terms, the expression becomes:
\[ \frac{47^{1/2}}{7^{-3}} = 47^{1/2} \cdot 7^{3} = 47^{1/2} \cdot 7^{3} \]
Thus, the rational exponent for the final expression is:
- For \(47\): \(1/2\)
- For \(7\): \(3\)
Your new combined expression could also be represented as follows, maintaining each term separately:
\[ \text{Final Result: } 47^{1/2} \cdot 7^{3} \]
In summary, the rational exponent for \(47\) is \(\frac{1}{2}\), and for \(7\) is \(3\).