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: $24=_{3}×3$.
- Variable part: $s_{3}=s×s×s$.

**12s⁴**: The numerical coefficient is 12, and the variable part is $s_{4}$.- Prime factorization of 12: $12=_{2}×3$.
- Variable part: $s_{4}=s×s×s×s$.

**18s**: The numerical coefficient is 18, and the variable part is $s$.- Prime factorization of 18: $18=2×_{2}$.
- 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: $_{3}×3$
- Prime factorization of 12: $_{2}×3$
- Prime factorization of 18: $2×_{2}$

**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 $_{1}$.
- For the factor 3: the lowest power is $_{1}$.

**Multiply the Lowest Powers Together**:- $GCF=_{1}×_{1}=2×3=6$

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.

$GCF of 24s_{3},12s_{4},and 18s=6s$

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.