The number of zeroes coming immediately after the decimal point in the value of `(0.2)^25` is : Given `log_10 2 = 0.30103`)

Options

A. 16
B. 17
C. 18
D. none of these

Explanation

To find the number of zeroes immediately after the decimal point in the value of \( (0.2)^{25} \), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting \( 0.2 \) in terms of powers of 10:
\[ 0.2 = \frac{2}{10} = 2 \times 10^{-1} \]
Thus,
\[ (0.2)^{25} = (2 \times 10^{-1})^{25} = 2^{25} \times (10^{-1})^{25} = 2^{25} \times 10^{-25} \]

### Step 2: Analyze the expression
Now, we can express \( (0.2)^{25} \) as:
\[ (0.2)^{25} = \frac{2^{25}}{10^{25}} \]
This shows that \( (0.2)^{25} \) is a fraction where the numerator is \( 2^{25} \) and the denominator is \( 10^{25} \).

### Step 3: Find the value of \( 2^{25} \)
To find the number of zeroes after the decimal point, we need to determine how many digits are in \( 2^{25} \). We can use logarithms to find this:
\[ \text{Number of digits in } n = \lfloor \log_{10} n \rfloor + 1 \]
So, we need to calculate \( \log_{10} (2^{25}) \):
\[ \log_{10} (2^{25}) = 25 \cdot \log_{10} 2 \]
Given \( \log_{10} 2 = 0.30103 \):
\[ \log_{10} (2^{25}) = 25 \cdot 0.30103 = 7.526 \]

### Step 4: Calculate the number of digits in \( 2^{25} \)
Now, we can find the number of digits in \( 2^{25} \):
\[ \text{Number of digits} = \lfloor 7.526 \rfloor + 1 = 7 + 1 = 8 \]

### Step 5: Determine the number of zeroes after the decimal
Since \( (0.2)^{25} = \frac{2^{25}}{10^{25}} \), the number of zeroes immediately after the decimal point can be calculated as:
\[ \text{Number of zeroes} = 25 - \text{Number of digits in } 2^{25} = 25 - 8 = 17 \]

### Final Answer
Thus, the number of zeroes coming immediately after the decimal point in the value of \( (0.2)^{25} \) is:
\[ \boxed{17} \]