Final answer:
To find the lowest common multiple (LCM) of the given expressions, we need to find the prime factors of each expression and determine the highest power of each prime factor. Once we have the prime factors and their highest powers, we can multiply them together to find the LCM.
Step-by-step explanation:
To find the lowest common multiple (LCM) of the given expressions, we need to find the prime factors of each expression and determine the highest power of each prime factor that appears in any of the given expressions. Here are the steps:
- Factorize each expression into its prime factors:
- 48x³y³z² = 2² × 2 × 2 × 2 × 3 × x × x × x × y × y × y × z × z
- 96x⁴y²z⁴ = 2³ × 2 × 2 × 2 × 2 × 3 × x × x × x × x × y × y × z × z × z × z
- 112x⁵y⁴z³ = 2 × 2 × 2 × 2 × 2 × 7 × x × x × x × x × x × y × y × y × y × z × z × z
- Determine the highest power of each prime factor:
- 2 appears with the highest power of 4
- 3 appears with the highest power of 1
- 7 appears with the highest power of 1
- x appears with the highest power of 5
- y appears with the highest power of 4
- z appears with the highest power of 4
- Multiply all the prime factors with their respective highest powers:
LCM = 2⁴ × 3 × 7 × x⁵ × y⁴ × z⁴ - Simplify the expression:
LCM = 16 × 3 × 7 × x⁵ × y⁴ × z⁴ - Final answer:
LCM = 336x⁵y⁴z⁴