Which of the following real numbers when simplified are neither terminating nor repeating decimal ?

Options

A. `sin75^(@)*cos75^(@)`
B. `log_(2)28`
C. `log_(3)5*log_(5)6`
D. `8^(-(log_(27)3))`

Answer

To determine which of the given real numbers are neither terminating nor repeating decimals, we will analyze each option step by step.

### Step 1: Analyze Option A
**Given:** \( \sin 75^\circ \cdot \cos 75^\circ \)

Using the double angle formula:
\[ \sin 2A = 2 \sin A \cos A \]
Let \( A = 75^\circ \):
\[ \sin 75^\circ \cdot \cos 75^\circ = \frac{1}{2} \sin 150^\circ \]
Now, we know:
\[ \sin 150^\circ = \sin(180^\circ - 30^\circ) = \sin 30^\circ = \frac{1}{2} \]
Thus,
\[ \sin 75^\circ \cdot \cos 75^\circ = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \]
**Conclusion:** \( \frac{1}{4} \) is a rational number, which can be expressed as \( 0.25 \), a terminating decimal.

### Step 2: Analyze Option B
**Given:** \( \log_2 28 \)

We can break this down using properties of logarithms:
\[ \log_2 28 = \log_2 (4 \cdot 7) = \log_2 4 + \log_2 7 \]
Since \( 4 = 2^2 \):
\[ \log_2 4 = 2 \]
Now we have:
\[ \log_2 28 = 2 + \log_2 7 \]
Using the change of base formula:
\[ \log_2 7 = \frac{\log_{10} 7}{\log_{10} 2} \]
Calculating \( \log_{10} 7 \) gives approximately \( 0.845 \) and \( \log_{10} 2 \) gives approximately \( 0.301 \):
\[ \log_2 7 \approx \frac{0.845}{0.301} \approx 2.807 \]
Thus:
\[ \log_2 28 \approx 2 + 2.807 = 4.807 \]
**Conclusion:** This value is neither terminating nor repeating.

### Step 3: Analyze Option C
**Given:** \( \log_5 3 \cdot \log_6 5 \)

Using the change of base formula:
\[ \log_5 3 = \frac{\log_{10} 3}{\log_{10} 5}, \quad \log_6 5 = \frac{\log_{10} 5}{\log_{10} 6} \]
Thus:
\[ \log_5 3 \cdot \log_6 5 = \frac{\log_{10} 3}{\log_{10} 5} \cdot \frac{\log_{10} 5}{\log_{10} 6} = \frac{\log_{10} 3}{\log_{10} 6} \]
Calculating \( \log_{10} 3 \approx 0.477 \) and \( \log_{10} 6 \approx 0.778 \):
\[ \frac{\log_{10} 3}{\log_{10} 6} \approx \frac{0.477}{0.778} \approx 0.613 \]
**Conclusion:** This value is also neither terminating nor repeating.

### Step 4: Analyze Option D
**Given:** \( 8^{-\log_{27} 3} \)

Using properties of logarithms:
\[ \log_{27} 3 = \frac{\log_{10} 3}{\log_{10} 27} = \frac{\log_{10} 3}{3 \log_{10} 3} = \frac{1}{3} \]
Thus:
\[ 8^{-\log_{27} 3} = 8^{-\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}} = \frac{1}{2} \]
**Conclusion:** \( \frac{1}{2} \) is a rational number and a terminating decimal.

### Final Conclusion
Out of the four options, the numbers that are neither terminating nor repeating decimals are:
- **Option B:** \( \log_2 28 \)
- **Option C:** \( \log_5 3 \cdot \log_6 5 \)