To simplify the given fraction \(\frac{3y^6}{4(y^{-1})^3}\), we can follow these steps:
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Simplify the denominator: \[ (y^{-1})^3 = y^{-3} \] So the denominator becomes: \[ 4y^{-3} \]
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Rewrite the fraction: Now, we have: \[ \frac{3y^6}{4y^{-3}} \]
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Use the property of exponents: We know that dividing by \(y^{-3}\) is the same as multiplying by \(y^3\): \[ \frac{3y^6}{4y^{-3}} = \frac{3y^6 \cdot y^3}{4} \] Combine the exponents of \(y\): \[ y^6 \cdot y^3 = y^{6 + 3} = y^9 \]
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Combine everything: Now the expression becomes: \[ \frac{3y^9}{4} \]
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Express in simplest form: The fraction \(\frac{3y^9}{4}\) is already in simplest form, using only positive exponents.
Thus, the final answer is: \[ \frac{3y^9}{4} \]