To simplify the expression \(\frac{8x^5y^7}{4x^3y^4}\) using algebraic rules for exponents, follow these steps:
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Simplify the coefficients: Divide the coefficients (numerical parts) first: \[ \frac{8}{4} = 2 \]
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Apply the quotient rule for exponents for \(x\): The quotient rule states that \(\frac{a^m}{a^n} = a^{m-n}\). Here, you will subtract the exponents of \(x\): \[ \frac{x^5}{x^3} = x^{5-3} = x^2 \]
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Apply the quotient rule for exponents for \(y\): Similarly, for \(y\): \[ \frac{y^7}{y^4} = y^{7-4} = y^3 \]
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Combine all parts together: Now combine the simplified coefficient and the terms with \(x\) and \(y\): \[ 2x^2y^3 \]
Thus, the simplified form of \(\frac{8x^5y^7}{4x^3y^4}\) is: \[ \boxed{2x^2y^3} \]