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

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 __

GPT-4o mini · Answer

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**.