When faced with complex algebraic expressions, one important concept to grasp is the Greatest Common Factor (GCF). The GCF is the largest expression that can divide each of the given terms without leaving a remainder. what-is-the-greatest-common-factor-of-42a5b3-35a3b4-and-42ab4-7ab3-6a4b-42a5b4-77a8b7.
This concept extends beyond just numerical values to include variables with exponents. In this article, we’ll explore the process of finding the GCF for the algebraic terms 42a⁵b³, 35a³b⁴, 42ab⁴, 7ab³, 6a⁴b, 42a⁵b⁴, and 77a⁸b⁷. By the end of this article, you’ll understand how to determine the GCF for any set of terms, answering the question: what is the greatest common factor of 42a⁵b³, 35a³b⁴, and 42ab⁴? 7ab³ 6a⁴b 42a⁵b⁴ 77a⁸b⁷.
1. Breaking Down Each Term into Its Prime Factors
To find the GCF, we first need to break down each term into its prime factors. This means expressing each coefficient as a product of prime numbers and writing out the variables with their exponents.
- 42a⁵b³:
- 42 can be factored as 2 × 3 × 7.
- a⁵ is already in its simplest form, and b³ is as well.
So, 42a⁵b³ = 2 × 3 × 7 × a⁵ × b³.
- 35a³b⁴:
- 35 can be factored as 5 × 7.
- a³ and b⁴ are already in their simplest forms.
So, 35a³b⁴ = 5 × 7 × a³ × b⁴.
- 42ab⁴:
- 42 as before is 2 × 3 × 7.
- a and b⁴ are in their simplest forms.
So, 42ab⁴ = 2 × 3 × 7 × a × b⁴.
- 7ab³:
- 7 is a prime number, so it stays as is.
- a and b³ are in their simplest forms.
So, 7ab³ = 7 × a × b³.
- 6a⁴b:
- 6 can be factored as 2 × 3.
- a⁴ and b are in their simplest forms.
So, 6a⁴b = 2 × 3 × a⁴ × b.
- 42a⁵b⁴:
- 42 as before is 2 × 3 × 7.
- a⁵ and b⁴ are in their simplest forms.
So, 42a⁵b⁴ = 2 × 3 × 7 × a⁵ × b⁴.
- 77a⁸b⁷:
- 77 can be factored as 7 × 11.
- a⁸ and b⁷ are in their simplest forms.
So, 77a⁸b⁷ = 7 × 11 × a⁸ × b⁷.
2. Identifying the Common Factors
Now that each term is broken down into its prime factors, the next step is to identify the factors common to all terms. The GCF must include only those factors that are present in each term.
- Numerical Coefficient:
- The only common numerical factor across all terms is 7. Other prime factors like 2, 3, 5, and 11 are not present in every term.
- Variable a:
- The smallest power of a that appears in every term is a. In the terms, a appears as a¹, a³, a⁴, a⁵, and a⁸. The GCF will use the smallest exponent, which is a.
- Variable b:
- The smallest power of b that appears in every term is b. In the terms, b appears as b¹, b³, b⁴, and b⁷. The GCF will use the smallest exponent, which is b.
what-is-the-greatest-common-factor-of-42a5b3-35a3b4-and-42ab4-7ab3-6a4b-42a5b4-77a8b7.Thus, for the expressions 42a⁵b³, 35a³b⁴, and 42ab⁴, the question arises: what is the greatest common factor of 42a⁵b³, 35a³b⁴, and 42ab⁴? 7ab³ 6a⁴b 42a⁵b⁴ 77a⁸b⁷? The answer, as we’ll see, is consistent across all these terms.
3. Writing the Greatest Common Factor
With the common factors identified, we can now write the GCF. It is the product of all the common factors we found:
GCF=7×a×b=7ab\text{GCF} = 7 \times a \times b = 7ab
So, the Greatest Common Factor of the terms 42a⁵b³, 35a³b⁴, 42ab⁴, 7ab³, 6a⁴b, 42a⁵b⁴, and 77a⁸b⁷ is 7ab. This answers our central question: what is the greatest common factor of 42a⁵b³, 35a³b⁴, and 42ab⁴? 7ab³ 6a⁴b 42a⁵b⁴ 77a⁸b⁷.
4. Understanding the Importance of the GCF
The GCF is a critical concept, especially when simplifying algebraic expressions or solving equations. By factoring out the GCF, you can reduce complex expressions into simpler forms, making them easier to work with. what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4? 7ab3 6a4b 42a5b4 77a8b7
It also plays a crucial role in operations like adding or subtracting fractions with algebraic terms, where a common denominator is needed.
5. Applying the GCF in Problem-Solving
Let’s apply the GCF we found in a real-world problem. Suppose you need to simplify an algebraic fraction or solve an equation involving the terms we discussed. Factoring out the GCF can simplify the problem significantly.
For instance, if you had the expression:
42a5b3+35a3b4+42ab47ab\frac{42a⁵b³ + 35a³b⁴ + 42ab⁴}{7ab}
You can factor out 7ab from the numerator:
7ab(6a4b2+5a2b3+6b3)7ab\frac{7ab(6a⁴b² + 5a²b³ + 6b³)}{7ab}
Simplifying by canceling the 7ab in the numerator and denominator, you get:
6a4b2+5a2b3+6b36a⁴b² + 5a²b³ + 6b³
This simplification makes the expression much more manageable.
6. Comparing GCF with Other Concepts
what-is-the-greatest-common-factor-of-42a5b3-35a3b4-and-42ab4-7ab3-6a4b-42a5b4-77a8b7The GCF is closely related to other concepts like the Least Common Multiple (LCM), but it’s essential to understand the distinction. While the GCF deals with the greatest common factor that divides all terms, the LCM concerns the smallest multiple that all terms can divide. In some problems, you might need to find both the GCF and the LCM to fully solve the problem.
7. Conclusion
what-is-the-greatest-common-factor-of-42a5b3-35a3b4-and-42ab4-7ab3-6a4b-42a5b4-77a8b7. The process of finding the Greatest Common Factor of algebraic expressions is a fundamental skill in algebra. By understanding how to break down terms into their prime factors, identify common factors, and then write the GCF, you can simplify expressions, solve equations, and approach more complex problems with confidence. what is the greatest common factor of 42a5b3, 35a3b4, and 42ab4? 7ab3 6a4b 42a5b4 77a8b7
The GCF of 42a⁵b³, 35a³b⁴, and 42ab⁴, as well as the other terms discussed, answers the central question: what is the greatest common factor of 42a⁵b³, 35a³b⁴, and 42ab⁴? 7ab³ 6a⁴b 42a⁵b⁴ 77a⁸b⁷. The answer is 7ab, which showcases the power of this concept in algebraic manipulation. Read More D2armorpicker.