HCF of `3^(3)xx5^(4)` and `3^(4)xx5^(2)` is _______________.

Answer

To find the HCF (Highest Common Factor) of the two numbers given in the question, we will follow these steps:

### Step 1: Identify the numbers
The two numbers we need to find the HCF for are:
1. \( 3^3 \times 5^4 \)
2. \( 3^4 \times 5^2 \)

### Step 2: Factor the numbers
We can express the numbers in their prime factorization:
- The first number is \( 3^3 \) and \( 5^4 \).
- The second number is \( 3^4 \) and \( 5^2 \).

### Step 3: Determine the HCF for each prime factor
For the prime factor \( 3 \):
- The powers are \( 3 \) (from the first number) and \( 4 \) (from the second number).
- The HCF is the minimum of these powers: \( \min(3, 4) = 3 \).

For the prime factor \( 5 \):
- The powers are \( 4 \) (from the first number) and \( 2 \) (from the second number).
- The HCF is the minimum of these powers: \( \min(4, 2) = 2 \).

### Step 4: Combine the HCFs of the prime factors
Now, we combine the HCFs of the prime factors:
\[ \text{HCF} = 3^{\min(3, 4)} \times 5^{\min(4, 2)} = 3^3 \times 5^2 \]

### Step 5: Write the final answer
Thus, the HCF of \( 3^3 \times 5^4 \) and \( 3^4 \times 5^2 \) is:
\[ 3^3 \times 5^2 \]

### Step 6: Calculate the numerical value (optional)
If needed, we can calculate the numerical value:
\[ 3^3 = 27 \quad \text{and} \quad 5^2 = 25 \]
So,
\[ 27 \times 25 = 675 \]

### Final Answer
The HCF of \( 3^3 \times 5^4 \) and \( 3^4 \times 5^2 \) is \( 3^3 \times 5^2 \) or numerically \( 675 \).

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