What is the HCF of `2^(3) xx 3^(4) and 2^(5) xx 3^(2)`?

Options

A. `2^(5) xx 3^(3)`
B. `2^(3) xx 3^(4)`
C. `2^(3) xx 3^(2)`
D. `2^(5) xx 3^(4)`

Answer

To find the Highest Common Factor (HCF) of the numbers \(2^3 \times 3^4\) and \(2^5 \times 3^2\), we can follow these steps:

### Step 1: Identify the prime factorization
The two numbers are:
1. \(2^3 \times 3^4\)
2. \(2^5 \times 3^2\)

### Step 2: List the powers of each prime factor
For the first number:
- The power of 2 is 3.
- The power of 3 is 4.

For the second number:
- The power of 2 is 5.
- The power of 3 is 2.

### Step 3: Determine the HCF for each prime factor
To find the HCF, we take the lowest power of each prime factor from both numbers.

- For the prime factor 2:
- The powers are 3 and 5. The minimum is \(3\).

- For the prime factor 3:
- The powers are 4 and 2. The minimum is \(2\).

### Step 4: Write the HCF using the lowest powers
Now, we can write the HCF as:
\[ \text{HCF} = 2^{\text{min}(3,5)} \times 3^{\text{min}(4,2)} = 2^3 \times 3^2 \]

### Step 5: Final Answer
Thus, the HCF of \(2^3 \times 3^4\) and \(2^5 \times 3^2\) is:
\[ \text{HCF} = 2^3 \times 3^2 \]