Question

Use the quotient property and negative exponent property to generate an equivalent expression for square root 49 over 7^-3. What will the rational exponent be for the new expression
The rational exponent for the new expression is __

Answer 1

To find an equivalent expression for \(\frac{\sqrt{49}}{7^{-3}}\), we'll start by simplifying both the numerator and the denominator.

1. **Calculate the square root in the numerator**:
\[
\sqrt{49} = 7
\]

2. **Now, substitute this into the expression**:
\[
\frac{\sqrt{49}}{7^{-3}} = \frac{7}{7^{-3}}
\]

3. **Use the quotient property of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\)**:
- Here, we have \(7^1\) in the numerator and \(7^{-3}\) in the denominator.
\[
\frac{7^1}{7^{-3}} = 7^{1 - (-3)} = 7^{1 + 3} = 7^{4}
\]

4. **Convert to rational exponents**:
- The expression \(7^{4}\) can also be represented as:
\[
7^{4} = 7^{\frac{4}{1}}
\]

So the rational exponent for the new expression is \(\frac{4}{1}\).

Thus, the final answer is:
The rational exponent for the new expression is **4**.