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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?(1 point)

1 answer

To simplify the expression \( \frac{49\sqrt{7}}{7^{-3}} \) using the Quotient Property and Negative Exponent Property, we follow these steps:

  1. Convert the Square Root to a Rational Exponent: The square root can be expressed with a rational exponent: \[ \sqrt{7} = 7^{1/2} \] Thus, we can rewrite \( 49\sqrt{7} \): \[ 49\sqrt{7} = 49 \cdot 7^{1/2} \]

  2. Simplify 49: We can express 49 as \( 7^2 \): \[ 49 = 7^2 \] Therefore: \[ 49\sqrt{7} = 7^2 \cdot 7^{1/2} = 7^{2 + 1/2} = 7^{5/2} \]

  3. Work on the Denominator: The denominator is \( 7^{-3} \). Using the Quotient Property \( \frac{a^m}{a^n} = a^{m-n} \), we can write: \[ \frac{7^{5/2}}{7^{-3}} = 7^{5/2 - (-3)} = 7^{5/2 + 3} \]

  4. Calculate the Exponent: Convert 3 to a fraction with a denominator of 2: \[ 3 = \frac{6}{2} \] Now, add the exponents: \[ 5/2 + 6/2 = \frac{5 + 6}{2} = \frac{11}{2} \]

  5. Final Expression: Therefore, the equivalent expression is: \[ 7^{11/2} \]

Thus, the rational exponent for the new expression is \( \frac{11}{2} \).