The number of zeros after decimal before the start of any significant digit in the number `N=(0.15)^(20)` are :

Options

A. 15
B. 16
C. 17
D. 18

Answer

To find the number of zeros after the decimal before the start of any significant digit in the number \( N = (0.15)^{20} \), we can follow these steps:

### Step 1: Express \( N \) in logarithmic form
We start with the expression:
\[ N = (0.15)^{20} \]
Taking the logarithm (base 10) of both sides, we have:
\[ \log_{10} N = \log_{10}((0.15)^{20}) \]

### Step 2: Use logarithmic properties
Using the property of logarithms that states \( \log(a^b) = b \cdot \log(a) \), we can rewrite the equation as:
\[ \log_{10} N = 20 \cdot \log_{10}(0.15) \]

### Step 3: Rewrite \( 0.15 \)
Next, we can express \( 0.15 \) as a fraction:
\[ 0.15 = \frac{15}{100} = \frac{15}{10^2} \]
Thus, we can write:
\[ \log_{10}(0.15) = \log_{10}\left(\frac{15}{10^2}\right) \]

### Step 4: Apply the quotient rule of logarithms
Using the property \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \), we have:
\[ \log_{10}(0.15) = \log_{10}(15) - \log_{10}(10^2) \]
Since \( \log_{10}(10^2) = 2 \):
\[ \log_{10}(0.15) = \log_{10}(15) - 2 \]

### Step 5: Break down \( \log_{10}(15) \)
We can express \( 15 \) as \( 3 \times 5 \):
\[ \log_{10}(15) = \log_{10}(3) + \log_{10}(5) \]

### Step 6: Substitute known logarithm values
Using known values:
\[ \log_{10}(3) \approx 0.477 \quad \text{and} \quad \log_{10}(5) \approx 0.699 \]
Thus:
\[ \log_{10}(15) \approx 0.477 + 0.699 = 1.176 \]

### Step 7: Substitute back into the logarithm equation
Now substituting back:
\[ \log_{10}(0.15) = 1.176 - 2 = -0.824 \]

### Step 8: Calculate \( \log_{10} N \)
Now substituting this back into our equation for \( \log_{10} N \):
\[ \log_{10} N = 20 \cdot (-0.824) = -16.48 \]

### Step 9: Interpret the result
The logarithmic value \( -16.48 \) indicates that:
\[ N = 10^{-16.48} \]
This means \( N \) is a very small number, specifically:
\[ N \approx 0.000000000000000031 \]
The number of zeros after the decimal before the first significant digit is the integer part of \( -16.48 \), which is \( 16 \).

### Final Answer
Thus, the number of zeros after the decimal before the start of any significant digit in \( N \) is:
\[ \boxed{16} \]