what is the greatest common factor of 60x4y7, 45x5y5, and 75x3y? 5xy 15x3y 45x3y5 75x5y7.When tackling algebraic expressions, one common question is: what is the greatest common factor of 60x⁴y⁷, 45x⁵y⁵, and 75x³y? 5xy 15x³y 45x³y⁵ 75x⁵y⁷. Understanding the Greatest Common Factor (GCF) is essential in simplifying these expressions and solving related problems. The GCF is the largest factor that divides all terms without leaving a remainder. In this article, we will break down each expression to determine the GCF, ensuring that you can answer the question effectively.
Analyzing the Expressions: Breaking Down the Factors
To answer the question what is the greatest common factor of 60x⁴y⁷, 45x⁵y⁵, and 75x³y? 5xy 15x³y 45x³y⁵ 75x⁵y⁷, we need to start by breaking down each expression into its prime factors:
- 60x⁴y⁷:
- Numerical factor: 60 can be factored into 2² × 3 × 5.
- Variable factor: x⁴y⁷ is already in its simplest form.
Therefore, 60x⁴y⁷ can be expressed as: 2² × 3 × 5 × x⁴ × y⁷.
- 45x⁵y⁵:
- Numerical factor: 45 can be factored into 3² × 5.
- Variable factor: x⁵y⁵ is already in its simplest form.
Therefore, 45x⁵y⁵ can be expressed as: 3² × 5 × x⁵ × y⁵.
- 75x³y:
- Numerical factor: 75 can be factored into 3 × 5².
- Variable factor: x³y is already in its simplest form.
Therefore, 75x³y can be expressed as: 3 × 5² × x³ × y.
Identifying the Common Factors
In the context of the query what is the greatest common factor of 60x⁴y⁷, 45x⁵y⁵, and 75x³y? 5xy 15x³y 45x³y⁵ 75x⁵y⁷, what is the greatest common factor of 60x4y7, 45x5y5, and 75x3y? 5xy 15x3y 45x3y5 75x5y7.
it is essential to identify the common factors from the prime factorization of each expression:
- Numerical factors:
- The common numerical factors are 3 and 5.
- Variable factors:
- The common variable factors depend on the lowest power of x and y present in each term.
- The lowest power of x in the given expressions is x³.
- The lowest power of y in the given expressions is y.
Thus, the common factors of the given expressions are:
- Numerical factor: 3 × 5 = 15
- Variable factor: x³y
Calculating the Greatest Common Factor (GCF)
After identifying the common factors, we calculate the GCF:
GCF=15×x3×y=15x3y\text{GCF} = 15 \times x³ \times y = 15x³y
Hence, when asked what is the greatest common factor of 60x⁴y⁷, 45x⁵y⁵, and 75x³y? 5xy 15x³y 45x³y⁵ 75x⁵y⁷, the correct answer is 15x³y.
Exploring Factor Combinations: 5xy, 15x³y, 45x³y⁵, 75x⁵y⁷
Let’s examine the provided factor combinations and their relevance to our calculated GCF:
- 5xy:
- This is a factor of all terms but not the greatest one.
- The correct GCF, as calculated, is 15x³y.
- 15x³y:
- This is the correct answer to the query what is the greatest common factor of 60x⁴y⁷, 45x⁵y⁵, and 75x³y? 5xy 15x³y 45x³y⁵ 75x⁵y⁷.
- 45x³y⁵:
- This factor is not common to all three expressions, as it includes higher powers of y.
- 75x⁵y⁷:
- This factor is larger than any common factor, making it incorrect for the GCF.
Real-World Applications of GCF
what is the greatest common factor of 60x4y7 45x5y5 and 75x3y? 5xy 15x3y 45x3y5 75x5y7. Understanding the GCF is crucial for various algebraic tasks. For example, if you need to simplify the ratio of two expressions, knowing the GCF is key:
- For instance, in simplifying 60x4y775x3y\frac{60x⁴y⁷}{75x³y}, the GCF is 15x³y, leading to: 60x4y775x3y=4xy65\frac{60x⁴y⁷}{75x³y} = \frac{4xy⁶}{5}
- This simplification is directly related to finding the GCF.
Conclusion: Answering the Key Question
In conclusion, when faced with the question what is the greatest common factor of 60x⁴y⁷, 45x⁵y⁵, and 75x³y? 5xy 15x³y 45x³y⁵ 75x⁵y⁷, the answer is 15x³y. what is the greatest common factor of 60x4y7 45x5y5 and 75x3y? 5xy 15x3y 45x3y5 75x5y7. The GCF plays a vital role in simplifying expressions and solving algebraic problems, making it an essential concept in mathematics. Read More D2armorpicker.