Problem 5 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of monomials. $$5 x^{2}, 12 y^{2}$$

Step-by-Step Solution

Answer

LCM(5x², 12y²) = 60x²y².

Step 1: Find the LCM of the numerical coefficients

To find the LCM of the numerical coefficients 5 and 12, we first list the multiples of each number. The first few multiples of 5 are 5, 10, 15, 20, 25, 30, 35, etc., and of 12 are 12, 24, 36, 48, etc. The smallest common multiple between these two lists is 60. Hence, LCM(5, 12) = 60.

Step 2: Find the LCM for the like variable parts

Since the monomials have different variables, they don’t have like terms. The LCM for the different variable parts (i.e., the variables themselves) will be the product of the different variable parts. Therefore, LCM of the variables is simply the product of them: x² * y² = x²y².

Step 3: Combine the results

Combine the results from Steps 1 and 2 to find the overall LCM of the monomials. We obtained 60 as the LCM of the coefficients and x²y² from the LCM of the variable parts. Hence, the LCM of the monomials 5x² and 12y² is 60x²y².

Key Concepts

Monomials

A monomial is a single term that comprises numeric coefficients and variable parts, which may include variables raised to whole number powers. It can be as simple as a single number like 5, or more complex, including both numbers and variables, such as \(5x^2\) or \(12y^2\). Monomials are the building blocks of more complex polynomials, and understanding them is crucial for various mathematical operations such as finding the least common multiple (LCM).
In the given exercise, each term (5x² and 12y²) is a monomial. The calculation of the LCM for such terms involves separate consideration of their numeric coefficients and their variable components. The term like \(5x^2\) displays the classic components of monomials: a coefficient '5' and a variable 'x' raised to the power of 2.

Numerical Coefficients

Numerical coefficients are the numbers that multiply the variables in a monomial. They give the monomial its size or scale, and when comparing monomials, these coefficients are crucial for calculations like finding the LCM.
In the exercise, the numerical coefficients are 5 and 12. To find the LCM of these coefficients, you need to consider their multiples and identify the smallest common one. This involves listing multiples until you reach a common value. In this case,

• The multiples of 5 include: 5, 10, 15, 20, 25, 30, etc.
• The multiples of 12 include: 12, 24, 36, 48, etc.

Here, the smallest common multiple is 60, making it the LCM of the coefficients.

Variables

Variables are symbols that represent numbers and can change in value. In a monomial, they can appear in conjunction with numerical coefficients and may be raised to any power.
In this exercise, the variables are 'x' and 'y', featured in the forms of \(x^2\) and \(y^2\). When dealing with variables that are part of monomials, finding the LCM requires logical handling, especially when variables are different.
Since these monomials have distinct variables, the LCM for these variable parts results from simply multiplying the variables together without any additional combination as they do not share common bases. Hence, the LCM of the variable parts in this example is \(x^{2}y^{2}\), maintaining both the variables and their respective powers.