To simplify the expression \( 49 \sqrt[7]{-3} \) using the Quotient Property and Negative Exponent Property, we will first rewrite the expression in terms of rational exponents.
- We can write \( 49 \) as \( 49 = 7^2 \).
- The expression \( \sqrt[7]{-3} \) can be written using a rational exponent as \( (-3)^{1/7} \).
Putting it all together:
\[ 49 \sqrt[7]{-3} = 7^2 \cdot (-3)^{1/7} \]
Next, we can apply the Quotient Property and Negative Exponent Property. Since there is no division or negative exponent directly involved in the original expression, we simply write it in terms of a product:
If we express it as:
\[ \frac{7^2}{(-3)^{-1/7}} \]
This would indicate carrying the value of \( -3 \) as a negative exponent, reflecting the concept of dividing by that term.
Now, the rational exponent for the new expression is:
\[ \frac{2}{1} + \frac{1}{7} \text{ or } 2 + \frac{1}{7} = \frac{14}{7} + \frac{1}{7} = \frac{15}{7} \]
So if we put the two expressions together, we can see that the rational exponent results to \( \frac{15}{7} \).
However, if your original intent was just to express \( 49 \sqrt[7]{-3} \), then you would have:
\[ 7^2 (-3)^{1/7} = 7^2 \cdot (-3)^{1/7} \]
In summary, applying the Quotient and Negative Exponent properties will give a final overall perspective of the combination but the rational exponent can be framed as:
The resulting rational exponent for the expression can be simplified as necessary, but traditionally remains as can be articulated clearly under these conversions.
The answer to the question about the rational exponent is \( \frac{15}{7} \) based on the manner of combining rational components if one needs this representation.
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