To simplify the expression \( 49\sqrt{7^{-3}} \) using the Quotient Property and the Negative Exponent Property, we first rewrite the square root in terms of an exponent.
The square root can be expressed as: \[ \sqrt{a} = a^{1/2} \] Thus, \[ \sqrt{7^{-3}} = (7^{-3})^{1/2} \]
Now apply the power of a power property (which states that \((a^m)^n = a^{m \cdot n}\)): \[ (7^{-3})^{1/2} = 7^{-3/2} \]
Now substitute this back into the expression \( 49\sqrt{7^{-3}} \): \[ 49\sqrt{7^{-3}} = 49 \cdot 7^{-3/2} \]
Next, we can express \( 49 \) as \( 7^2 \): \[ 49 = 7^2 \]
Now substituting this into the expression gives: \[ 7^2 \cdot 7^{-3/2} \]
Using the Product Property of exponents (which states \( a^m \cdot a^n = a^{m+n} \)): \[ 7^2 \cdot 7^{-3/2} = 7^{2 - 3/2} \]
To combine \( 2 \) and \(-3/2\), we convert \( 2 \) to a fraction: \[ 2 = \frac{4}{2} \] Thus, \[ 2 - \frac{3}{2} = \frac{4}{2} - \frac{3}{2} = \frac{1}{2} \]
So the exponent simplifies to: \[ 7^{\frac{1}{2}} \]
Therefore, the expression \( 49\sqrt{7^{-3}} \) simplifies to: \[ 7^{\frac{1}{2}} \]
The rational exponent for the new expression is: \[ \frac{1}{2} \]