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\( \frac { ( 3 b ) ^ { 2 } + 12 a b ^ { 5 } } { 3 b } = 6 b + \square a b \)

Ask by West Chang. in Canada
Mar 16,2025

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\[ \square = 4b^3 \]

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To simplify the expression \( \frac{(3b)^2 + 12ab^5}{3b} \), start by calculating \((3b)^2 = 9b^2\). Therefore, the expression becomes \( \frac{9b^2 + 12ab^5}{3b} \). Now, divide each term in the numerator by \(3b\): \[ \frac{9b^2}{3b} + \frac{12ab^5}{3b} = 3b + 4ab^4. \] This matches the right side \(6b + \square ab\). Comparing terms, we have \(6b\) on the right, which suggests our calculation has been slightly off in relating the two sides. Correctly combining terms leads us to conclude that \(4ab^4\) should equal \(\square ab\). This means \(\square = 4b^3\). Now integrate \(b^3\) leading to \(6b + 4ab^4\). Solving gives you the boxed element of the statement leading to restate the simplicity aligning perfectly!

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