Find the HCF of `(2^(5) xx 5^(2) xx 11^(1)), (3^(2) xx 5^(3) xx 11^(2))` and `(2^(4) xx 3^(6) xx 5^(1) xx 7^4)`

Answer

To find the HCF (Highest Common Factor) of the three numbers given, we will break down the problem step by step.

### Step 1: Identify the prime factorization of each number
1. The first number is \(2^5 \times 5^2 \times 11^1\).
2. The second number is \(3^2 \times 5^3 \times 11^2\).
3. The third number is \(2^4 \times 3^6 \times 5^1 \times 7^4\).

### Step 2: List the prime factors
The prime factors involved in these numbers are \(2\), \(3\), \(5\), \(7\), and \(11\).

### Step 3: Find the lowest power of each prime factor common to all three numbers
- For \(2\):
- First number: \(2^5\)
- Second number: \(2^0\) (since \(2\) is not a factor)
- Third number: \(2^4\)

The lowest power is \(2^0\).

- For \(3\):
- First number: \(3^0\) (since \(3\) is not a factor)
- Second number: \(3^2\)
- Third number: \(3^6\)

The lowest power is \(3^0\).

- For \(5\):
- First number: \(5^2\)
- Second number: \(5^3\)
- Third number: \(5^1\)

The lowest power is \(5^1\).

- For \(7\):
- First number: \(7^0\) (since \(7\) is not a factor)
- Second number: \(7^0\) (since \(7\) is not a factor)
- Third number: \(7^4\)

The lowest power is \(7^0\).

- For \(11\):
- First number: \(11^1\)
- Second number: \(11^2\)
- Third number: \(11^0\) (since \(11\) is not a factor)

The lowest power is \(11^0\).

### Step 4: Combine the lowest powers
Now, we combine the lowest powers of all the prime factors:
\[ HCF = 2^0 \times 3^0 \times 5^1 \times 7^0 \times 11^0 \]

### Step 5: Simplify the expression
Since any number raised to the power of zero is \(1\), we have:
\[ HCF = 1 \times 1 \times 5^1 \times 1 \times 1 = 5 \]

### Final Answer
The HCF of the three numbers is **5**.