Find The Greatest Common Factor Of These Two Expressions: 12 V 8 U 5 W 6 12v^8u^5w^6 12 V 8 U 5 W 6 And 16 V 7 W 3 16v^7w^3 16 V 7 W 3

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Introduction

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In algebra, the greatest common factor (GCF) of two or more expressions is the largest expression that divides each of the given expressions without leaving a remainder. Finding the GCF of algebraic expressions is an essential skill in mathematics, as it helps us simplify complex expressions and solve equations. In this article, we will learn how to find the GCF of two given expressions: $12v^8u^5w^6$ and $16v^7w^3$.

Understanding the Concept of GCF

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The GCF of two or more expressions is the product of the common factors of the expressions. To find the GCF, we need to identify the common factors of the expressions and multiply them together. The common factors are the factors that appear in both expressions.

Step 1: Factorize the Expressions

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To find the GCF, we need to factorize the given expressions. Factorizing an expression involves breaking it down into its prime factors.

Factorizing $12v^8u^5w^6$

The expression $12v^8u^5w^6$ can be factorized as follows:

• $12 = 2^2 \times 3$
• $v^8 = v^4 \times v^4$
• $u^5 = u^5$
• $w^6 = w^3 \times w^3$

So, the factorized form of $12v^8u^5w^6$ is:

$$2^2 \times 3 \times v^4 \times v^4 \times u^5 \times w^3 \times w^3$$

Factorizing $16v^7w^3$

The expression $16v^7w^3$ can be factorized as follows:

• $16 = 2^4$
• $v^7 = v^7$
• $w^3 = w^3$

So, the factorized form of $16v^7w^3$ is:

$$2^4 \times v^7 \times w^3$$

Step 2: Identify the Common Factors

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Now that we have factorized the expressions, we can identify the common factors. The common factors are the factors that appear in both expressions.

In this case, the common factors are:

• $2^2$
• $v^4$
• $w^3$

Step 3: Multiply the Common Factors

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To find the GCF, we need to multiply the common factors together.

The GCF of $12v^8u^5w^6$ and $16v^7w^3$ is:

$$2^2 \times v^4 \times w^3$$

Simplifying the GCF

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We can simplify the GCF by combining the powers of the common factors.

The simplified GCF is:

$$4v^4w^3$$

Conclusion

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In this article, we learned how to find the greatest common factor (GCF) of two algebraic expressions: $12v^8u^5w^6$ and $16v^7w^3$. We factorized the expressions, identified the common factors, and multiplied them together to find the GCF. The GCF of the two expressions is $4v^4w^3$.

Example Problems

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Problem 1

Find the GCF of $18x^5y^3$ and $24x^4y^2$.

Solution

To find the GCF, we need to factorize the expressions.

The expression $18x^5y^3$ can be factorized as follows:

• $18 = 2 \times 3^2$
• $x^5 = x^5$
• $y^3 = y^3$

So, the factorized form of $18x^5y^3$ is:

$$2 \times 3^2 \times x^5 \times y^3$$

The expression $24x^4y^2$ can be factorized as follows:

• $24 = 2^3 \times 3$
• $x^4 = x^4$
• $y^2 = y^2$

So, the factorized form of $24x^4y^2$ is:

$$2^3 \times 3 \times x^4 \times y^2$$

The common factors are:

• $2$
• $3$
• $x^4$
• $y^2$

The GCF is:

$$2 \times 3 \times x^4 \times y^2$$

The simplified GCF is:

$$6x^4y^2$$

Problem 2

Find the GCF of $20x^6y^4$ and $30x^5y^3$.

Solution

To find the GCF, we need to factorize the expressions.

The expression $20x^6y^4$ can be factorized as follows:

• $20 = 2^2 \times 5$
• $x^6 = x^6$
• $y^4 = y^4$

So, the factorized form of $20x^6y^4$ is:

$$2^2 \times 5 \times x^6 \times y^4$$

The expression $30x^5y^3$ can be factorized as follows:

• $30 = 2 \times 3 \times 5$
• $x^5 = x^5$
• $y^3 = y^3$

So, the factorized form of $30x^5y^3$ is:

$$2 \times 3 \times 5 \times x^5 \times y^3$$

The common factors are:

• $5$
• $x^5$
• $y^3$

The GCF is:

$$5 \times x^5 \times y^3$$

The simplified GCF is:

$$5x^5y^3$$

Final Thoughts

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Finding the greatest common factor (GCF) of algebraic expressions is an essential skill in mathematics. By factorizing the expressions, identifying the common factors, and multiplying them together, we can find the GCF. The GCF can be simplified by combining the powers of the common factors. In this article, we learned how to find the GCF of two algebraic expressions and solved example problems to reinforce our understanding.

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Frequently Asked Questions

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Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) of two or more expressions is the largest expression that divides each of the given expressions without leaving a remainder.

Q: How do I find the GCF of two expressions?

A: To find the GCF, you need to factorize the expressions, identify the common factors, and multiply them together.

Q: What are the steps to find the GCF?

A: The steps to find the GCF are:

1. Factorize the expressions.
2. Identify the common factors.
3. Multiply the common factors together.

Q: Can I simplify the GCF?

A: Yes, you can simplify the GCF by combining the powers of the common factors.

Q: What if the expressions have different variables?

A: If the expressions have different variables, you can ignore the variables that are not common to both expressions.

Q: Can I find the GCF of more than two expressions?

A: Yes, you can find the GCF of more than two expressions by following the same steps.

Q: How do I know if the GCF is correct?

A: To check if the GCF is correct, you can divide each of the original expressions by the GCF and see if the result is a whole number.

Q: What if the GCF is not a whole number?

A: If the GCF is not a whole number, it means that the expressions do not have a common factor.

Q: Can I use the GCF to simplify equations?

A: Yes, you can use the GCF to simplify equations by factoring out the GCF from each term.

Q: What are some common mistakes to avoid when finding the GCF?

A: Some common mistakes to avoid when finding the GCF include:

• Not factorizing the expressions correctly.
• Not identifying the common factors correctly.
• Not multiplying the common factors together correctly.

Q: How can I practice finding the GCF?

A: You can practice finding the GCF by working through example problems and exercises.

Q: What are some real-world applications of the GCF?

A: The GCF has many real-world applications, including:

• Simplifying equations in physics and engineering.
• Finding the greatest common divisor in computer science.
• Solving problems in finance and economics.

Additional Resources

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If you want to learn more about the greatest common factor (GCF), here are some additional resources:

• Khan Academy: Greatest Common Factor (GCF)
• Mathway: Greatest Common Factor (GCF)
• Wolfram Alpha: Greatest Common Factor (GCF)

Conclusion

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The greatest common factor (GCF) is an essential concept in mathematics that has many real-world applications. By understanding how to find the GCF, you can simplify equations, solve problems, and make informed decisions. Remember to factorize the expressions, identify the common factors, and multiply them together to find the GCF. With practice and patience, you can become proficient in finding the GCF and apply it to a wide range of problems.

Final Thoughts

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Finding the greatest common factor (GCF) is a valuable skill that can be applied to many areas of mathematics and real-world problems. By mastering the GCF, you can simplify equations, solve problems, and make informed decisions. Remember to practice regularly and seek help when needed to become proficient in finding the GCF.