To simplify the expression \( \frac{x^3 y^{-6}}{x^7 y^{-10}} \), we need to consider the properties of exponents.
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Negative Exponent Property: This property states that \( a^{-n} = \frac{1}{a^n} \). In this expression, we have \( y^{-6} \) and \( y^{-10} \). If we apply the negative exponent property first, we can rewrite both \( y^{-6} \) and \( y^{-10} \) in terms of positive exponents. However, this isn't necessary to begin simplifying the expression.
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Quotient of Powers Property: This property states that \( \frac{a^m}{a^n} = a^{m-n} \). We can apply this property to both the \( x \) and \( y \) terms in the expression.
Given the current structure of the expression, the most effective property to apply first is the Quotient of Powers Property to simplify the bases of \( x \) and \( y \):
- For \( x \): \( \frac{x^3}{x^7} = x^{3-7} = x^{-4} \)
- For \( y \): \( \frac{y^{-6}}{y^{-10}} = y^{-6 - (-10)} = y^{4} \)
Thus, the Quotient of Powers Property is the appropriate choice to use first when simplifying this expression.
So the answer is: Quotient of Powers Property.