Find the LCM of [tex]$3 x^2 y^3$[/tex] and [tex]$10 x y^4$[/tex].
Answer (2)
The LCM of 3 x 2 y 3 and 10 x y 4 is calculated by finding the LCM of the coefficients and the highest powers of the variables. The coefficients are 3 and 10, with an LCM of 30. Therefore, the final LCM is 30 x 2 y 4 .
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Find the LCM of the coefficients: LCM(3, 10) = 30.
Identify the highest power of x: x 2 .
Identify the highest power of y: y 4 .
Combine these to form the LCM of the terms: 30 x 2 y 4 .
Explanation
Understanding the Problem We are asked to find the least common multiple (LCM) of 3 x 2 y 3 and 10 x y 4 . The LCM is the smallest expression that both terms divide into evenly.
LCM of Coefficients To find the LCM, we need to consider the LCM of the coefficients and the highest powers of each variable present in the given terms.
First, let's find the LCM of the coefficients 3 and 10. The LCM of 3 and 10 is 30, since 3 and 10 have no common factors other than 1.
Highest Powers of Variables Next, we consider the variables. We have x 2 in the first term and x in the second term. The highest power of x is x 2 .
We have y 3 in the first term and y 4 in the second term. The highest power of y is y 4 .
Combining the Results Now, we multiply the LCM of the coefficients by the highest powers of each variable to get the LCM of the two terms: 30 x 2 y 4 .
Final Answer Therefore, the LCM of 3 x 2 y 3 and 10 x y 4 is 30 x 2 y 4 .
Examples
Understanding LCM is crucial in many real-world scenarios. For instance, when scheduling events that occur at different intervals, like planning when two buses on different routes will arrive at the same stop simultaneously. Or, in manufacturing, determining the minimum number of products to order so that you have enough of each component when they are supplied in different quantities per package. The LCM helps optimize these processes, ensuring efficiency and minimizing waste.
