Use quotient property and negative Exponent Property to generate an equivalent expression for (\sqrt(49))/(7^(-3)). What will the rational exponent be for the new expression?

Question

7^

Answer 1

To simplify the expression \(\frac{\sqrt{49}}{7^{-3}}\) using the quotient property and negative exponent property, we start by simplifying the components.

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

2. The expression now becomes: \[ \frac{7}{7^{-3}} \]

3. Applying the quotient property of exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\): \[ \frac{7^1}{7^{-3}} = 7^{1 - (-3)} = 7^{1 + 3} = 7^4 \]

Now, \(7^4\) is the simplified expression.

To express \(7^4\) in terms of rational exponents, we can write: \[ 7^4 = 7^{\frac{4}{1}} \]

So, the rational exponent for the new expression is: \[ \frac{4}{1} \] and the final answer is: \[ 7^4 \]