Find the HCF and LCM of 6x²y²z, 24x⁴y⁴z², 300x³y³z³.

Points to Remember:

• Finding the Highest Common Factor (HCF) involves identifying the greatest common divisor of given numbers or expressions.
• Finding the Least Common Multiple (LCM) involves identifying the smallest common multiple of given numbers or expressions.
• For algebraic expressions, HCF and LCM involve considering the common factors of the variables and coefficients.

Introduction:

The Highest Common Factor (HCF) and Least Common Multiple (LCM) are fundamental concepts in number theory and algebra. The HCF represents the largest number that divides all the given numbers without leaving a remainder. The LCM represents the smallest number that is a multiple of all the given numbers. Finding the HCF and LCM of algebraic expressions like polynomials involves identifying the common factors of the variables and their coefficients. This question requires us to find the HCF and LCM of three algebraic expressions: 6x²y²z, 24x⁴y⁴z², and 300x³y³z³. This will involve factoring each expression and identifying common and unique factors.

Body:

1. Finding the HCF:

To find the HCF, we first find the prime factorization of the coefficients and identify the lowest power of each common variable:

• 6 = 2 × 3
• 24 = 2³ × 3
• 300 = 2² × 3 × 5²

The common factors of the coefficients are 2 and 3. The lowest power of ‘x’ is x², the lowest power of ‘y’ is y², and the lowest power of ‘z’ is z.

Therefore, the HCF of 6x²y²z, 24x⁴y⁴z², and 300x³y³z³ is: 2 × 3 × x² × y² × z = 6x²y²z

2. Finding the LCM:

To find the LCM, we identify the highest power of each variable and the highest power of each prime factor present in the coefficients:

• Highest power of 2: 2³ = 8
• Highest power of 3: 3
• Highest power of 5: 5² = 25
• Highest power of x: x⁴
• Highest power of y: y⁴
• Highest power of z: z³

Therefore, the LCM of 6x²y²z, 24x⁴y⁴z², and 300x³y³z³ is: 2³ × 3 × 5² × x⁴ × y⁴ × z³ = 3000x⁴y⁴z³

Conclusion:

In summary, the HCF of 6x²y²z, 24x⁴y⁴z², and 300x³y³z³ is 6x²y²z, and the LCM is 3000x⁴y⁴z³. Finding the HCF and LCM of algebraic expressions is a crucial skill in simplifying expressions, solving equations, and working with fractions involving polynomials. Understanding these concepts is essential for further advancements in algebra and related mathematical fields. This process highlights the importance of systematic factorization and careful consideration of both numerical coefficients and variable powers. The application of these methods ensures accuracy and efficiency in solving similar problems. A strong foundation in these fundamental concepts promotes a deeper understanding of more complex mathematical concepts.