Find the HCF of `36x^(3)y^(2)and24x^(4)y`.

Answer

To find the HCF (Highest Common Factor) of the expressions \(36x^3y^2\) and \(24x^4y\), we will follow these steps:

### Step 1: Factor the coefficients
First, we need to factor the numerical coefficients (36 and 24).

- The prime factorization of 36:
\[ 36 = 2^2 \times 3^2 \]

- The prime factorization of 24:
\[ 24 = 2^3 \times 3^1 \]

### Step 2: Write down the variables
Next, we will write down the variables in each expression:
- For \(36x^3y^2\), we have:
\[ x^3 \text{ and } y^2 \]

- For \(24x^4y\), we have:
\[ x^4 \text{ and } y^1 \]

### Step 3: Find the HCF of the coefficients
Now, we will find the HCF of the coefficients \(36\) and \(24\):
- For the prime factor \(2\), the minimum power is \(2\) (from \(2^2\) and \(2^3\)).
- For the prime factor \(3\), the minimum power is \(1\) (from \(3^2\) and \(3^1\)).

Thus, the HCF of the coefficients is:
\[ HCF = 2^2 \times 3^1 = 4 \times 3 = 12 \]

### Step 4: Find the HCF of the variables
Next, we will find the HCF of the variables:
- For \(x\), the minimum power is \(3\) (from \(x^3\) and \(x^4\)).
- For \(y\), the minimum power is \(1\) (from \(y^2\) and \(y^1\)).

Thus, the HCF of the variables is:
\[ HCF = x^3 \times y^1 = x^3y \]

### Step 5: Combine the results
Finally, we combine the HCF of the coefficients and the variables to get the overall HCF:
\[ HCF = 12 \times x^3y = 12x^3y \]

### Final Answer
The HCF of \(36x^3y^2\) and \(24x^4y\) is:
\[ \boxed{12x^3y} \]