To simplify the expression \(\frac{3x^3y^{-4}}{12x^{7}y^{4}}\) using the properties of integer exponents, follow these steps:
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Simplify the coefficients: \[ \frac{3}{12} = \frac{1}{4} \]
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Simplify the \(x\) terms using the property \(a^m / a^n = a^{m-n}\): \[ \frac{x^3}{x^7} = x^{3-7} = x^{-4} \]
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Simplify the \(y\) terms: \[ \frac{y^{-4}}{y^{4}} = y^{-4-4} = y^{-8} \]
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Combine the results: Putting all these together, we get: \[ \frac{3x^3y^{-4}}{12x^{7}y^{4}} = \frac{1}{4} x^{-4} y^{-8} \]
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Write the expression using positive exponents: To express it with positive exponents, we can rewrite \(x^{-4}\) and \(y^{-8}\) as: \[ \frac{1}{4x^4y^8} \]
Thus, the equivalent algebraic expression is: \[ \frac{1}{4x^4y^8} \]