What is HCF of Monomials? | Simple Explanation for Class 7

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What is the Highest Common Factor (HCF) of Monomials?

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Highest Common Factor (HCF) of monomials is the largest monomial that divides each of the given monomials exactly. Think of it as finding the biggest common 'building block' that fits into all of them.

Simple Example

Quick Example

Imagine you have two friends, Rohan and Priya, who love collecting cricket cards. Rohan has 6 Virat Kohli cards and 9 Rohit Sharma cards. Priya has 12 Virat Kohli cards and 15 Rohit Sharma cards. If we want to find the HCF of their Virat Kohli cards (6 and 12), it's 6. This is similar to finding the HCF of the number parts of monomials.

Worked Example

Step-by-Step

Let's find the HCF of 12x^2y and 18xy^3.

STEP 1: Find the HCF of the numerical coefficients (numbers). The numbers are 12 and 18.
Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 18: 1, 2, 3, 6, 9, 18
The HCF of 12 and 18 is 6.
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STEP 2: Find the HCF of the variable parts. For each common variable, take the lowest power.
Common variable 'x': We have x^2 and x^1. The lowest power is x^1 (or just x).
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STEP 3: Common variable 'y': We have y^1 and y^3. The lowest power is y^1 (or just y).
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STEP 4: Multiply the HCF of the numbers by the HCF of each common variable.
HCF = 6 * x * y
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Answer: The HCF of 12x^2y and 18xy^3 is 6xy.

Why It Matters

Understanding HCF of monomials helps you simplify complex algebraic expressions, which is super important in fields like Physics to solve equations about motion, or in Computer Science for optimizing code. Engineers use these concepts to design efficient systems, from building bridges to creating new apps!

Common Mistakes

MISTAKE: Taking the highest power of variables instead of the lowest. For example, for x^2 and x^3, students might take x^3. | CORRECTION: Always take the lowest power for each common variable when finding HCF, because the HCF must divide *all* terms.

MISTAKE: Forgetting to include variables that appear in all monomials. For example, for 6x^2y and 9xy^2, students might find HCF as 3x^2y^2. | CORRECTION: Only include variables that are common to ALL monomials, and take their lowest power. The HCF is 3xy.

MISTAKE: Not finding the HCF of the numerical coefficients correctly. For example, for 10x and 15y, taking 5 as HCF of numbers, but ignoring the variables. | CORRECTION: First, find the HCF of the numbers. Then, for each common variable, find its lowest power. If no variables are common, the HCF of variables is 1. If no numbers are common, the HCF of numbers is 1. Here, HCF of 10 and 15 is 5. No common variables. So HCF is 5.

Practice Questions

Try It Yourself

QUESTION: Find the HCF of 4a^2b and 6ab. | ANSWER: 2ab

QUESTION: What is the HCF of 15p^3q^2 and 20p^2q^4? | ANSWER: 5p^2q^2

QUESTION: Determine the HCF of 24x^3yz, 36x^2y^2, and 60xy^3z^2. | ANSWER: 12xy

MCQ

Quick Quiz

What is the HCF of 8m^4n and 12m^2n^3?

4m^4n^3

4m^2n

2m^2n

12m^4n^3

The Correct Answer Is:

The HCF of 8 and 12 is 4. For 'm', the lowest power is m^2. For 'n', the lowest power is n^1. So, the HCF is 4m^2n.

Real World Connection

In the Real World

Imagine you're a software developer creating an app for managing inventory at a local kirana store. You might use HCF of monomials when optimizing storage space for different products, where 'x' could be the number of large boxes and 'y' the number of small packets. Finding the HCF helps you identify the most efficient common unit for packaging.

Key Vocabulary

Key Terms

MONOMIAL: An algebraic expression with only one term, like 5x or 3y^2 | COEFFICIENT: The numerical part of a monomial, like 5 in 5x | VARIABLE: A letter representing an unknown value, like x or y | FACTOR: A number or expression that divides another number or expression exactly | HIGHEST COMMON FACTOR (HCF): The largest factor common to two or more numbers or expressions

What's Next

What to Learn Next

Great job understanding HCF of monomials! Next, you can explore 'Factorization of Algebraic Expressions'. This concept builds directly on HCF, helping you break down complex expressions into simpler parts, which is a fundamental skill in algebra.