Use the Quotient Property and Negative Exponent Property to generate an equivalent expression for radical 49√7^−3

Question

. What will the rational exponent be for the new expression?(1 point)
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Answer 1

To simplify the expression \( \sqrt[49]{7^{-3}} \) using the Quotient Property and Negative Exponent Property, we start with the negative exponent property:

1. Start with the expression: \[ \sqrt[49]{7^{-3}} \]

2. Convert the radical to a rational exponent: The nth root can be expressed as a fractional exponent. Therefore, we rewrite the expression: \[ 7^{-3} = \frac{1}{7^3} \]

3. Now substitute back into the expression: \[ \sqrt[49]{7^{-3}} = (7^{-3})^{\frac{1}{49}} \]

4. Apply the power of a power property (multiply the exponents): \[ (7^{-3})^{\frac{1}{49}} = 7^{-3 \cdot \frac{1}{49}} = 7^{-\frac{3}{49}} \]

The simplified expression is:

\[ 7^{-\frac{3}{49}} \]

Now, to identify the rational exponent:

The rational exponent for the new expression is:

\[ -\frac{3}{49} \]

Thus, the equivalent expression is \( 7^{-\frac{3}{49}} \) with the rational exponent being \( -\frac{3}{49} \).