(a) Calculate the lowest common multiple of \( 24 a^{2} b^{3} \) and \( 16 a b^{2} \).

To find the **lowest common multiple (LCM)** of the two expressions \( 24 a^{2} b^{3} \) and \( 16 a b^{2} \), follow these steps: 1. **Factor Each Expression:** - **First Expression: \( 24 a^{2} b^{3} \)** - **Numerical part:** \( 24 = 2^{3} \times 3 \) - **Variables:** - \( a^{2} \) - \( b^{3} \) - **Complete Factorization:** \( 2^{3} \times 3 \times a^{2} \times b^{3} \) - **Second Expression: \( 16 a b^{2} \)** - **Numerical part:** \( 16 = 2^{4} \) - **Variables:** - \( a \) - \( b^{2} \) - **Complete Factorization:** \( 2^{4} \times a \times b^{2} \) 2. **Determine the Highest Powers of Each Factor:** - **For the numerical factors:** Choose the highest exponent of each prime number. - **2:** The highest power between \( 2^{3} \) and \( 2^{4} \) is \( 2^{4} \). - **3:** Only appears in the first expression as \( 3^{1} \). - **For the variables:** Choose the highest exponent for each variable. - **\( a \):** The highest power between \( a^{2} \) and \( a \) is \( a^{2} \). - **\( b \):** The highest power between \( b^{3} \) and \( b^{2} \) is \( b^{3} \). 3. **Combine the Highest Powers to Form the LCM:** \[ \text{LCM} = 2^{4} \times 3 \times a^{2} \times b^{3} \] 4. **Simplify the Numerical Part:** \[ 2^{4} \times 3 = 16 \times 3 = 48 \] 5. **Final LCM Expression:** \[ \text{LCM} = 48 a^{2} b^{3} \] **Answer:** The lowest common multiple is \( 48\, a^{2}\, b^{3} \).