Total number of digits in the product of `(4)^(1111) xx (5)^(2222)` is :

Options

A. 3333
B. 2223
C. 2222
D. can't be determined

Answer

To find the total number of digits in the product \( (4)^{1111} \times (5)^{2222} \), we can follow these steps:

### Step 1: Rewrite the base
We start by rewriting \( 4 \) as \( 2^2 \):
\[ (4)^{1111} = (2^2)^{1111} = 2^{2222} \]
So, we can rewrite the original expression:
\[ (4)^{1111} \times (5)^{2222} = 2^{2222} \times 5^{2222} \]

### Step 2: Combine the powers
Using the property of exponents \( a^m \times b^m = (a \times b)^m \), we can combine the terms:
\[ 2^{2222} \times 5^{2222} = (2 \times 5)^{2222} = 10^{2222} \]

### Step 3: Determine the number of digits
To find the total number of digits in \( 10^{2222} \), we use the formula for the number of digits in a number \( n \):
\[ \text{Number of digits} = \lfloor \log_{10} n \rfloor + 1 \]
For \( n = 10^{2222} \):
\[ \text{Number of digits} = \lfloor \log_{10} (10^{2222}) \rfloor + 1 = \lfloor 2222 \rfloor + 1 = 2222 + 1 = 2223 \]

### Final Answer
Thus, the total number of digits in the product \( (4)^{1111} \times (5)^{2222} \) is \( \boxed{2223} \).

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