Find the GCF and the factored form of 14x⁵×21x³

To find the Greatest Common Factor (GCF) and the factored form of the expression \( 14x^5 \times 21x^3 \), let's break it down step-by-step.

### Step 1: Find the GCF of the Coefficients
The coefficients are 14 and 21.

- The prime factorization of 14 is \( 2 \times 7 \).
- The prime factorization of 21 is \( 3 \times 7 \).

The GCF is the product of all common prime factors:
\[ \text{GCF of 14 and 21} = 7 \]

### Step 2: Find the GCF of the Variables
The variables are \( x^5 \) and \( x^3 \).

The GCF is the lowest power of \( x \) that appears in both terms:
\[ \text{GCF of } x^5 \text{ and } x^3 = x^3 \]

### Step 3: Combine the GCFs
Combine the GCF of the coefficients and the GCF of the variables:
\[ \text{GCF} = 7x^3 \]

### Step 4: Write the Expression in Factored Form
To express \( 14x^5 \times 21x^3 \) in factored form, we divide each term by the GCF and then express the result as a product:

\[ 14x^5 \times 21x^3 = 14x^5 \cdot 21x^3 = (7x^3) \cdot (2x^2) \cdot (3x^0) = 7x^3 \cdot (14x^2 \cdot 21) \]

This simplifies to:
\[ 14x^5 \times 21x^3 = 7x^3 \cdot (2 \cdot 3 \cdot x^2) = 7x^3 \cdot 6x^2 = 7x^3 \cdot (6x^2 \cdot 3x^0) = 7x^3 \cdot 6x^5 \]

So the factored form is:
\[ 7x^3 \cdot (2x^2 \cdot 3x^0) \]

### Conclusion
- **GCF:** \( 7x^3 \)
- **Factored Form:** \( 7x^3 (2x^2 \cdot 3x^0) \)

Would you like more details or have any questions?

Here are some related questions you might find useful:
1. What is the GCF of \( 18x^4 \) and \( 27x^6 \)?
2. How do you find the GCF of two algebraic expressions?
3. Can you explain the process of factoring a polynomial?
4. What is the difference between the GCF and the LCM?
5. How do you determine the prime factorization of a number?
6. What is the factored form of \( 36x^7 \times 24x^4 \)?
7. How can you simplify \( 14x^5 \times 21x^3 \) using the distributive property?
8. How do you factor expressions involving higher powers of variables?

**Tip:** When factoring expressions, always start by identifying and factoring out the greatest common factor to simplify the problem.

Discover how to find the Greatest Common Factor (GCF) and the factored form of the algebraic expression 14x⁵ × 21x³. This problem covers key algebraic concepts including GCF determination, factoring polynomials, and simplifying algebraic expressions. Suitable for students in Grades 7-9.

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