(a) Calculate the lowest common multiple of and
.

Ask by Sherman Curry. in South Africa

Jan 27,2025

Question

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

UpStudy AI · Accepted Answer

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 $$ 48\, a^{2}\, b^{3} $$.
The lowest common multiple is $$ 48\, a^{2}\, b^{3} $$.

(a) Calculate the lowest common multiple of and
.

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