What is the Greatest Common Factor of 24s³, 12s⁴, and 18s? A Step-by-Step Guide

what is the greatest common factor of 24s3 12s4 and 18s?. Understanding how to find the greatest common factor (GCF) is a fundamental skill in algebra that allows us to simplify expressions and solve problems more effectively. This article will address the specific question: What is the greatest common factor of 24s³, 12s⁴, and 18s? We will provide a step-by-step guide to finding the GCF of these algebraic expressions, breaking down the process to ensure a clear understanding.

1. What is the Greatest Common Factor?

Before answering “what is the greatest common factor of 24s³, 12s⁴, and 18s?”, it’s essential to understand what the GCF is. The greatest common factor (GCF) of two or more numbers or algebraic expressions is the largest number or expression that divides all of them without leaving a remainder. For algebraic expressions, the GCF includes both numerical coefficients and variables, where each variable is raised to the lowest power common to all expressions.

For instance, the GCF of 8 and 12 is 4, as 4 is the largest number that divides both 8 and 12 without leaving a remainder. In algebra, the GCF also includes variables and their exponents. Let’s apply this concept to our given expressions.

2. Analyzing the Expressions: 24s³, 12s⁴, and 18s

To find out what is the greatest common factor of 24s³, 12s⁴, and 18s, we need to break down each expression into its prime factors. This involves separating the numerical coefficients from the variables and expressing each part in its prime factorized form.

24s³: The numerical coefficient is 24, and the variable part is $s^{3}$ .
- Prime factorization of 24: $2^3 \times 3$ .
- Variable part: $s3=s×s×ss^3 = s \times s \times s$ .
12s⁴: The numerical coefficient is 12, and the variable part is $s^{4}$ .
- Prime factorization of 12: $2^2 \times 3$ .
- Variable part: $s4=s×s×s×ss^4 = s \times s \times s \times s$ .
18s: The numerical coefficient is 18, and the variable part is $s$ .
- Prime factorization of 18: $\times 3^2$ .
- Variable part: $s = s$ .

3. Determining the GCF of the Numerical Coefficients: 24, 12, and 18

To find the greatest common factor of 24s³, 12s⁴, and 18s, identify the largest number dividing 24, 12, and 18.

Prime factorization of 24: $23×32^3 \times 3$
Prime factorization of 12: $22×32^2 \times 3$
Prime factorization of 18: $\times 3^2$

Step-by-Step Process to Find the GCF:

Identify the Common Prime Factors: The common prime factors among 24, 12, and 18 are 2 and 3.
Choose the Lowest Power of Each Common Prime Factor:
- For the factor 2: the lowest power is $2^1$ .
- For the factor 3: the lowest power is $3^1$ .
Multiply the Lowest Powers Together:
- $2^1 \times 3^1 = 2 \times 3 = 6$

Thus, the GCF of the numerical coefficients 24, 12, and 18 is 6.

4. Finding the GCF of the Variable Parts: s³, s⁴, and s

Next, to determine what is the greatest common factor of 24s³, 12s⁴, and 18s, we need to find the GCF of the variable parts, which are $s^3$ , $s^4$ , and $s$ . The GCF of variables is found by taking the variable raised to the lowest power present in all the expressions.

The powers of $s$ in the expressions are 3, 4, and 1.
The lowest power is $s^1$ .

Therefore, the GCF of $s^3$ , $s^4$ , and $s$ is $s^1 = s$ .

5. Combining the GCF of the Numerical and Variable Parts

With GCFs of 6 and $s$ , the final answer for 24s³, 12s⁴, and 18s is 6s.

$18s=6s\text{GCF of 24s}^3, \text{12s}^4, \text{and 18s} = 6s$

Therefore, the greatest common factor of the algebraic expressions 24s³, 12s⁴, and 18s is 6s.

6. Applications of Finding the GCF in Algebra

Understanding what is the greatest common factor of 24s³, 12s⁴, and 18s and how to calculate it is crucial in algebra and higher mathematics. The GCF allows for the simplification of expressions, factoring polynomials, and solving equations. Here are a few practical applications:

Simplifying Fractions: In algebraic fractions, finding the GCF helps reduce them to their simplest form by dividing the numerator and the denominator by their GCF.
Factoring Polynomials: In polynomial expressions, the GCF helps in factoring out common terms, which is essential for solving quadratic and higher-order equations. what is the greatest common factor of 24s3 12s4 and 18s?
Solving Real-World Problems: In practical scenarios involving ratios, rates, and proportions, the GCF is often used to simplify calculations and find the correct solutions.

7. Conclusion

We answered “what is the greatest common factor of 24s³, 12s⁴, and 18s?” with a clear, step-by-step method. Breaking down both numerical coefficients and variable parts and identifying their common factors is crucial. The GCF of 24s³, 12s⁴, and 18s is 6s, combining the numerical GCF (6) with the variable part $s$ .

Mastering the process of finding the GCF helps in simplifying complex expressions, solving equations, and understanding foundational concepts in mathematics. Read More D2armorpicker.