Product of divisors of 7056 is :

Options

A. `(84)^(48)`
B. `(84)^(44)`
C. `(84)^(45)`
D. None of these

Answer

To find the product of the divisors of the number 7056, we can follow these steps:

### Step 1: Find the prime factorization of 7056
To find the prime factorization, we divide the number by the smallest prime numbers until we reach 1.

1. 7056 ÷ 2 = 3528
2. 3528 ÷ 2 = 1764
3. 1764 ÷ 2 = 882
4. 882 ÷ 2 = 441
5. 441 ÷ 3 = 147
6. 147 ÷ 3 = 49
7. 49 ÷ 7 = 7
8. 7 ÷ 7 = 1

Thus, the prime factorization of 7056 is:
\[ 7056 = 2^4 \times 3^2 \times 7^2 \]

### Step 2: Determine the number of divisors
The formula to find the number of divisors from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times ... \times p_n^{e_n} \) is:
\[ (e_1 + 1)(e_2 + 1)(e_3 + 1)...(e_n + 1) \]

For \( 7056 = 2^4 \times 3^2 \times 7^2 \):
- \( e_1 = 4 \) (for 2)
- \( e_2 = 2 \) (for 3)
- \( e_3 = 2 \) (for 7)

So, the number of divisors \( d \) is:
\[ d = (4 + 1)(2 + 1)(2 + 1) = 5 \times 3 \times 3 = 45 \]

### Step 3: Calculate the product of the divisors
The product of the divisors of a number \( n \) is given by the formula:
\[ n^{d/2} \]
where \( d \) is the number of divisors.

From Step 2, we found \( d = 45 \). Therefore, the product of the divisors of 7056 is:
\[ 7056^{45/2} = 7056^{22.5} \]

### Step 4: Final Result
Thus, the product of the divisors of 7056 is:
\[ 7056^{22.5} \]