Which of the following is/are true ?

Options

A. number of digits in `8^(12)5^(35)` is 35
B. number of digits in `8^(12)5^(35)` is 36
C. number of zeroes after decimal before a significant figures starts in `((8)/(27))^(20)` is 10
D. number of zeroes after decimal before a significant figure starts in `((8)/(27))^(20)` is 11

Explanation

To solve the problem step by step, we will analyze both parts of the question regarding the number of digits in \(8^{12} \times 5^{35}\) and the number of zeros after the decimal in \(\frac{8}{27^{20}}\).

### Part 1: Number of Digits in \(8^{12} \times 5^{35}\)

1. **Define the expression**:
Let \( n_1 = 8^{12} \times 5^{35} \).

2. **Take logarithm (base 10)**:
\[ \log_{10} n_1 = \log_{10} (8^{12} \times 5^{35}) \]

3. **Use logarithm properties**:
\[ \log_{10} n_1 = 12 \log_{10} 8 + 35 \log_{10} 5 \]

4. **Express \(8\) and \(5\) in terms of logarithms**:
\[ \log_{10} 8 = \log_{10} (2^3) = 3 \log_{10} 2 \]
\[ \log_{10} 5 \text{ is already known.} \]

5. **Substitute values**:
Using \(\log_{10} 2 \approx 0.3010\) and \(\log_{10} 5 \approx 0.6990\):
\[ \log_{10} n_1 = 12 \times (3 \times 0.3010) + 35 \times 0.6990 \]
\[ = 12 \times 0.9030 + 24.465 \]
\[ = 10.836 + 24.465 = 35.301 \]

6. **Calculate the number of digits**:
The number of digits \(d\) in a number \(n\) can be found using the formula:
\[ d = \lfloor \log_{10} n \rfloor + 1 \]
Thus,
\[ d = \lfloor 35.301 \rfloor + 1 = 35 + 1 = 36 \]

### Part 2: Number of Zeros After Decimal in \(\frac{8}{27^{20}}\)

1. **Define the expression**:
Let \( n_2 = \frac{8}{27^{20}} \).

2. **Take logarithm (base 10)**:
\[ \log_{10} n_2 = \log_{10} 8 - \log_{10} (27^{20}) \]

3. **Use logarithm properties**:
\[ \log_{10} n_2 = \log_{10} 8 - 20 \log_{10} 27 \]

4. **Express \(27\) in terms of logarithms**:
\[ \log_{10} 27 = \log_{10} (3^3) = 3 \log_{10} 3 \]

5. **Substitute values**:
Using \(\log_{10} 8 \approx 0.9030\) and \(\log_{10} 3 \approx 0.4771\):
\[ \log_{10} n_2 = 0.9030 - 20 \times (3 \times 0.4771) \]
\[ = 0.9030 - 20 \times 1.4313 \]
\[ = 0.9030 - 28.626 = -27.723 \]

6. **Calculate the number of zeros after the decimal**:
The number of zeros after the decimal before a significant figure starts can be found using:
\[ \text{Number of zeros} = \lfloor -\log_{10} n_2 \rfloor \]
Thus,
\[ \text{Number of zeros} = \lfloor 27.723 \rfloor = 27 \]

### Final Answers:
- The number of digits in \(8^{12} \times 5^{35}\) is **36**.
- The number of zeros after the decimal before a significant figure starts in \(\frac{8}{27^{20}}\) is **27**.