The HCF and LCM of `36x^(2)y^(3) and 32x^(3)y^(2)` are __________ and ___________ respectively.

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the polynomials \(36x^2y^3\) and \(32x^3y^2\), we will follow these steps:

### Step 1: Prime Factorization
First, we need to perform the prime factorization of both terms.

- For \(36x^2y^3\):
- \(36 = 6 \times 6 = 2^2 \times 3^2\)
- Therefore, \(36x^2y^3 = 2^2 \times 3^2 \times x^2 \times y^3\)

- For \(32x^3y^2\):
- \(32 = 2^5\)
- Therefore, \(32x^3y^2 = 2^5 \times x^3 \times y^2\)

### Step 2: Finding HCF
To find the HCF, we take the lowest power of each common factor.

- For \(2\): The powers are \(2^2\) and \(2^5\). The lowest power is \(2^2\).
- For \(3\): The powers are \(3^2\) and \(3^0\) (since \(32x^3y^2\) has no factor of \(3\)). The lowest power is \(3^0\).
- For \(x\): The powers are \(x^2\) and \(x^3\). The lowest power is \(x^2\).
- For \(y\): The powers are \(y^3\) and \(y^2\). The lowest power is \(y^2\).

Thus, the HCF is:
\[ HCF = 2^2 \times 3^0 \times x^2 \times y^2 = 4x^2y^2 \]

### Step 3: Finding LCM
To find the LCM, we take the highest power of each factor.

- For \(2\): The powers are \(2^2\) and \(2^5\). The highest power is \(2^5\).
- For \(3\): The powers are \(3^2\) and \(3^0\). The highest power is \(3^2\).
- For \(x\): The powers are \(x^2\) and \(x^3\). The highest power is \(x^3\).
- For \(y\): The powers are \(y^3\) and \(y^2\). The highest power is \(y^3\).

Thus, the LCM is:
\[ LCM = 2^5 \times 3^2 \times x^3 \times y^3 \]

Calculating the numerical part:
\[ 2^5 = 32, \quad 3^2 = 9 \]
So,
\[ LCM = 32 \times 9 \times x^3 \times y^3 = 288x^3y^3 \]

### Final Answer
The HCF and LCM of \(36x^2y^3\) and \(32x^3y^2\) are:
\[ \text{HCF} = 4x^2y^2 \quad \text{and} \quad \text{LCM} = 288x^3y^3 \]

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