Question

Use the Quotient Property and Negative Exponent Property to generate an equivalent expression for radical 49√7^−3
. 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 \( -\frac{3}{49} \).