To simplify the expression \(49\sqrt{7}^{-3}\) using the Quotient Property and Negative Exponent Property, we first need to rewrite the square root with a rational exponent.
The square root can be expressed as: \[ \sqrt{7} = 7^{1/2} \]
Thus, we can rewrite \(49\) as \(49 = 7^2\), leading to: \[ 49\sqrt{7}^{-3} = (7^2)(7^{1/2})^{-3} \]
Now, applying the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we can simplify \((7^{1/2})^{-3}\): \[ (7^{1/2})^{-3} = 7^{(1/2)(-3)} = 7^{-3/2} \]
Now we can write the entire expression: \[ 49\sqrt{7}^{-3} = 7^2 \cdot 7^{-3/2} \]
Using the Quotient Property, which states \(a^m \cdot a^n = a^{m+n}\), we combine the exponents: \[ 7^2 \cdot 7^{-3/2} = 7^{2 - 3/2} \]
To combine these exponents, convert \(2\) to a fraction: \[ 2 = \frac{4}{2} \] So we get: \[ 2 - 3/2 = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \]
Thus, the expression simplifies to: \[ 49\sqrt{7}^{-3} = 7^{1/2} \]
The final equivalent expression is \(7^{1/2}\), and the rational exponent for the new expression is: \[ \frac{1}{2} \]