Find the numbers of zeroes between the decimal point and first significant digit of `(0.036)^16`, where `log 2=0.301` and `log 3=0.477`.

Answer

To find the number of zeros between the decimal point and the first significant digit of \( (0.036)^{16} \), we can follow these steps:

### Step 1: Rewrite the expression
Let \( y = (0.036)^{16} \).

### Step 2: Take the logarithm
Taking logarithm on both sides, we have:
\[ \log y = 16 \log(0.036) \]

### Step 3: Express \( 0.036 \) in terms of fractions
We can express \( 0.036 \) as:
\[ 0.036 = \frac{36}{1000} = \frac{36}{10^3} \]
Thus,
\[ \log(0.036) = \log(36) - \log(1000) = \log(36) - 3 \]

### Step 4: Substitute back into the logarithm equation
Substituting this back, we get:
\[ \log y = 16 (\log(36) - 3) = 16 \log(36) - 48 \]

### Step 5: Break down \( \log(36) \)
Next, we can express \( 36 \) as \( 9 \times 4 \):
\[ \log(36) = \log(9) + \log(4) \]
Since \( 9 = 3^2 \) and \( 4 = 2^2 \), we can write:
\[ \log(9) = 2 \log(3) \quad \text{and} \quad \log(4) = 2 \log(2) \]
Thus,
\[ \log(36) = 2 \log(3) + 2 \log(2) = 2(\log(3) + \log(2)) \]

### Step 6: Substitute the values of \( \log(2) \) and \( \log(3) \)
Using the given values \( \log(2) = 0.301 \) and \( \log(3) = 0.477 \):
\[ \log(36) = 2(0.477 + 0.301) = 2(0.778) = 1.556 \]

### Step 7: Substitute back into the logarithm equation
Now substituting \( \log(36) \) back into the equation for \( \log y \):
\[ \log y = 16(1.556) - 48 \]
Calculating \( 16 \times 1.556 \):
\[ 16 \times 1.556 = 24.896 \]
Thus,
\[ \log y = 24.896 - 48 = -23.104 \]

### Step 8: Determine the number of zeros
The logarithm \( \log y = -23.104 \) indicates that \( y \) is a very small number. The number of zeros between the decimal point and the first significant digit can be found by taking the integer part of \( -\log y \):
\[ -\log y = 23.104 \]
The integer part of \( 23.104 \) is \( 23 \). Therefore, the number of zeros between the decimal point and the first significant digit is \( 23 \).

### Final Answer
The number of zeros between the decimal point and the first significant digit of \( (0.036)^{16} \) is **23**.

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