Problem 27 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers or monomials. $$75 n^{2}, 25 n^{4}$$

Step-by-Step Solution

Answer

The LCM of \(75n^2\) and \(25n^4\) is \(75n^4\).

Step 1: Understand the Problem

We need to find the least common multiple (LCM) of the monomials \(75n^2\) and \(25n^4\). The LCM is the smallest expression that both monomials divide into without a remainder.

Step 2: Factor Each Monomial Completely

First, factor \(75n^2\) and \(25n^4\):- \(75n^2 = 3 imes 5^2 imes n^2\)- \(25n^4 = 5^2 imes n^4\)

Step 3: Determine the LCM

To find the LCM, take the highest power of each factor present in either monomial:- For the constant terms, the higher power is \(5^2\).- For the variable \(n\), the higher power is \(n^4\).Therefore, the LCM is \(3 imes 5^2 imes n^4 = 75n^4\).

Key Concepts

Understanding Monomials

Monomials are algebraic expressions that consist of only one term. This includes a constant, a variable, or a product of constants and variables. For instance, in the problem given, we have the monomials \(75n^2\) and \(25n^4\). Here, \(n\) is the variable, and each monomial is a term involving a constant (75 or 25) multiplied by powers of \(n\).
Understanding monomials is crucial because it allows us to perform operations like addition, subtraction, or multiplication easily. In problems like finding the Least Common Multiple (LCM) of monomials, recognizing each component of the monomials is necessary for proper factoring.
It is also essential to recognize that the operations on a monomial strictly involve multiplication, which differentiates them from polynomials, having more than one term.

The Process of Factoring

Factoring is breaking down a number or expression into a product of its simplest components, known as factors. In mathematics, when you factor a number or a monomial, you're identifying the prime numbers and variables that multiply together to form the original expression.

• The factor tree method is often used for numbers, breaking them down progressively into prime factors.
• For the monomial \(75n^2\), we factor it as: \(75 = 3 \times 5^2\) and \(n^2\), resulting in the complete factorization \(3 \times 5^2 \times n^2\).
• Similarly, we factor \(25n^4\) as: \(25 = 5^2\) and \(n^4\), which gives \(5^2 \times n^4\).

Factoring sets the stage for finding the Least Common Multiple by identifying the base components involved. It simplifies the process of comparing and combining the factors from each monomial.

LCM Calculation

LCM, or Least Common Multiple, is an essential concept especially when working with monomials and polynomials. The LCM of two monomials is the smallest monomial that each of the given monomials divides without leaving a remainder.
To calculate the LCM:

• Consider each distinct factor that appears in either of the original monomials.
• For each distinct factor, use the highest power of that factor found in any of the given monomials.

For the problem with monomials \(75n^2\) and \(25n^4\):

• We compare and use the highest power of the factor \(5\), which is \(5^2\).
• We also compare the power of the variable \(n\) and take the highest, which is \(n^4\).
  

This leads to the LCM: \(3 \times 5^2 \times n^4 = 75n^4\).
Understanding LCM helps in solving equations, simplifying expressions, and working with fractional monomials more effectively.