Find the least common multiple of 9b^4 and 12y^3 square

To find the least common multiple (LCM) of \(9b^4\) and \(12y^3\), we need to follow these steps:<br /><br />1. **Factorize each term into its prime factors:**<br /> - \(9b^4 = 3^2 \cdot b^4\)<br /> - \(12y^3 = 2^2 \cdot 3 \cdot y^3\)<br /><br />2. **Identify the highest powers of all prime factors and variables involved:**<br /> - The prime factors are \(2\) and \(3\).<br /> - The variable factors are \(b\) and \(y\).<br /><br />3. **Take the highest power of each prime factor and variable:**<br /> - For \(2\), the highest power is \(2^2\).<br /> - For \(3\), the highest power is \(3^2\).<br /> - For \(b\), the highest power is \(b^4\).<br /> - For \(y\), the highest power is \(y^3\).<br /><br />4. **Multiply these highest powers together to get the LCM:**<br /> \[<br /> \text{LCM} = 2^2 \cdot 3^2 \cdot b^4 \cdot y^3<br /> \]<br /><br />5. **Calculate the product:**<br /> \[<br /> 2^2 = 4<br /> \]<br /> \[<br /> 3^2 = 9<br /> \]<br /> \[<br /> 4 \cdot 9 = 36<br /> \]<br /> \[<br /> 36 \cdot b^4 \cdot y^3 = 36b^4y^3<br /> \]<br /><br />Therefore, the least common multiple of \(9b^4\) and \(12y^3\) is:<br />\[<br />\boxed{36b^4y^3}<br />\]