Find the lowest common multiple of $8 x^2 y$ and $12 x y^2 z$.
A. $24 x^2 y^2 z$
B. $96 x^3 y^3 z$
C. $96 x^2 y^2 z$
D. $48 x^2 y^2 z$
Answer (1)
Find the LCM of the coefficients: LCM(8, 12) = 24.
Identify the highest powers of each variable: x 2 , y 2 , and z .
Multiply the LCM of the coefficients by the highest powers of each variable.
The lowest common multiple is 24 x 2 y 2 z .
Explanation
Understanding the Problem We are asked to find the lowest common multiple (LCM) of two expressions: 8 x 2 y and 12 x y 2 z . The LCM is the smallest expression that is divisible by both given expressions.
Finding the LCM To find the LCM, we need to find the LCM of the coefficients and the highest power of each variable present in the expressions.
LCM of Coefficients First, let's find the LCM of the coefficients 8 and 12. The LCM of 8 and 12 is 24.
Highest Powers of Variables Now, let's find the highest power of each variable:
The highest power of x is x 2 .
The highest power of y is y 2 .
The highest power of z is z 1 (or simply z ).
Combining the Results Finally, we multiply the LCM of the coefficients by the highest powers of each variable to get the LCM of the two expressions: 24 x 2 y 2 z .
Final Answer Therefore, the lowest common multiple of 8 x 2 y and 12 x y 2 z is 24 x 2 y 2 z .
Examples
In scheduling tasks, if one task repeats every 8 x 2 y time units and another repeats every 12 x y 2 z time units, the LCM 24 x 2 y 2 z represents the shortest time interval after which both tasks will occur simultaneously. This concept is crucial in optimizing resource allocation and preventing conflicts in various operational scenarios.
