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?

Answer 1

To simplify the expression \(\frac{\sqrt{49}}{7^{-3}}\), we will use the quotient property and negative exponent property.

1. Calculate \(\sqrt{49}\): \[ \sqrt{49} = 7 \]

2. Rewrite the expression: \[ \frac{\sqrt{49}}{7^{-3}} = \frac{7}{7^{-3}} \]

3. Using the quotient property: The quotient property states that \(\frac{a^m}{a^n} = a^{m-n}\). Here, we can apply this: \[ \frac{7^{1}}{7^{-3}} = 7^{1 - (-3)} = 7^{1 + 3} = 7^{4} \]

4. Convert the expression to a rational exponent: The expression \(7^{4}\) can be expressed with a rational exponent as: \[ 7^{4} = 7^{\frac{4}{1}} \]

Therefore, the equivalent expression for \(\frac{\sqrt{49}}{7^{-3}}\) is \(7^{4}\), and the rational exponent is \(\frac{4}{1}\).