Find the LCM of [tex]$22 a^3 b^3 c^4$[/tex] and [tex]$55 a b c$[/tex].
Answer (2)
To find the LCM of 22 a 3 b 3 c 4 and 55 ab c , we first calculate the LCM of the coefficients, resulting in 110 . Then, we identify the highest powers of each variable, which are a 3 , b 3 , and c 4 . The final LCM is 110 a 3 b 3 c 4 .
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Find the prime factorization of the coefficients: 22 = 2 × 11 and 55 = 5 × 11 .
Determine the LCM of the coefficients: lcm ( 22 , 55 ) = 2 × 5 × 11 = 110 .
Identify the highest powers of each variable: a 3 , b 3 , and c 4 .
Multiply the LCM of the coefficients by the highest powers of each variable: 110 a 3 b 3 c 4 .
Explanation
Problem Analysis We are asked to find the least common multiple (LCM) of two expressions: 22 a 3 b 3 c 4 and 55 ab c .
Objective To find the LCM of these two expressions, we need to find the LCM of the coefficients and the highest power of each variable present in either expression.
Prime Factorization of Coefficients First, let's find the prime factorization of the coefficients 22 and 55: 22 = 2 × 11 55 = 5 × 11
LCM of Coefficients Now, we determine the LCM of the coefficients: lcm ( 22 , 55 ) = 2 × 5 × 11 = 110
Highest Powers of Variables Next, we find the highest power of each variable:
For a , the highest power is a 3 .
For b , the highest power is b 3 .
For c , the highest power is c 4 .
LCM of Expressions Finally, we multiply the LCM of the coefficients by the highest powers of each variable to find the LCM of the expressions: 110 a 3 b 3 c 4
Final Answer Therefore, the LCM of 22 a 3 b 3 c 4 and 55 ab c is 110 a 3 b 3 c 4 .
Examples
In real life, the LCM is useful when scheduling events that occur at different intervals. For example, if one task needs to be done every 22 a 3 b 3 c 4 days and another task needs to be done every 55 ab c days, the LCM tells you when both tasks will need to be done on the same day. This helps in coordinating schedules and resources efficiently.
