Find the highest common factor(HCF) of (12xpower3y)power2 and 216xpower4y

Let's find the **Highest Common Factor (HCF)** of the two expressions:

### Given:
1. \((12x^3y)^2\)
2. \(216x^4y\)

### Step 1: Expand the powers in the first expression.

\[
(12x^3y)^2 = 12^2 \cdot (x^3)^2 \cdot y^2 = 144 \cdot x^6 \cdot y^2
\]

So, the expanded forms of the expressions are:

1. \(144x^6y^2\)
2. \(216x^4y\)

### Step 2: Factorize each expression.

#### Expression 1: \(144x^6y^2\)
- \(144 = 2^4 \cdot 3^2\)
- \(x^6\) is already factored.
- \(y^2\) is already factored.

So, \(144x^6y^2 = 2^4 \cdot 3^2 \cdot x^6 \cdot y^2\).

#### Expression 2: \(216x^4y\)
- \(216 = 2^3 \cdot 3^3\)
- \(x^4\) is already factored.
- \(y\) is already factored.

So, \(216x^4y = 2^3 \cdot 3^3 \cdot x^4 \cdot y\).

### Step 3: Identify the common factors.

Now, find the lowest powers of the common factors in both expressions:
- For \(2\), the lowest power is \(2^3\).
- For \(3\), the lowest power is \(3^2\).
- For \(x\), the lowest power is \(x^4\).
- For \(y\), the lowest power is \(y\).

### Step 4: Write the HCF.

The HCF is the product of the lowest powers of the common factors:
\[
HCF = 2^3 \cdot 3^2 \cdot x^4 \cdot y = 8 \cdot 9 \cdot x^4 \cdot y = 72x^4y
\]

Thus, the **HCF** of \((12x^3y)^2\) and \(216x^4y\) is:

\[
\boxed{72x^4y}
\]

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Would you like further details or have any questions?

### 5 Related Questions:
1. What is the Least Common Multiple (LCM) of these two expressions?
2. How do we find the HCF of polynomials?
3. Can you explain how the power of variables affects the HCF?
4. What is the HCF of numerical constants like 144 and 216?
5. How would this change if there were additional variables in the expressions?

**Tip:** Always factorize each term completely when finding the HCF!

Find the highest common factor (HCF) of (12x^3y)^2 and 216x^4y.

Learn how to find the Highest Common Factor (HCF) of the algebraic expressions (12x^3y)^2 and 216x^4y with a detailed step-by-step solution. This problem involves key algebraic concepts like exponents, factorization, and finding the HCF by comparing the lowest powers of common factors. Suitable for students in Grades 9-11.

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