If a number is in the form of `8^(10) xx 9^(7) xx 7^(8)`, find the total number of prime factors of the given number

Options

A. 52
B. 560
C. 3360
D. 25

Answer

To find the total number of prime factors of the number given in the form \( 8^{10} \times 9^{7} \times 7^{8} \), we will follow these steps:

### Step 1: Rewrite the numbers in terms of their prime factors
- The number \( 8 \) can be expressed as \( 2^3 \).
- The number \( 9 \) can be expressed as \( 3^2 \).
- The number \( 7 \) is already a prime number.

Thus, we can rewrite the expression:
\[ 8^{10} = (2^3)^{10} = 2^{30} \]
\[ 9^{7} = (3^2)^{7} = 3^{14} \]
\[ 7^{8} = 7^{8} \]

### Step 2: Combine the prime factorization
Now, we can combine these results:
\[ 8^{10} \times 9^{7} \times 7^{8} = 2^{30} \times 3^{14} \times 7^{8} \]

### Step 3: Identify the prime factors and their powers
From the combined expression, we have:
- The prime factor \( 2 \) with an exponent of \( 30 \)
- The prime factor \( 3 \) with an exponent of \( 14 \)
- The prime factor \( 7 \) with an exponent of \( 8 \)

### Step 4: Calculate the total number of prime factors
To find the total number of prime factors, we add the exponents of the prime factors:
\[ 30 + 14 + 8 = 52 \]

### Final Answer
The total number of prime factors of the given number is \( 52 \).

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