Solve the followings :
The value of the expression
`((0.3)^(1//3).(1/27)^(1//4) . (9)^(1//6).(0.81)^(2//3))/((0.9)^(2//3) . (3)^(-1//2) . (1/3)^(-2). (243)^(-1//4))` is :

Options

A. `0.3`
B. `0.9`
C. `1.27`
D. `0.09`

Answer

To solve the expression

\[ \frac{(0.3)^{1/3} \cdot (1/27)^{1/4} \cdot (9)^{1/6} \cdot (0.81)^{2/3}}{(0.9)^{2/3} \cdot (3)^{-1/2} \cdot (1/3)^{-2} \cdot (243)^{-1/4}} \]

we will simplify both the numerator and the denominator step by step.

### Step 1: Simplify the Numerator

1. **Calculate each term in the numerator:**
- \( (0.3)^{1/3} = (3/10)^{1/3} = \frac{3^{1/3}}{10^{1/3}} \)
- \( (1/27)^{1/4} = (27^{-1})^{1/4} = 27^{-1/4} = (3^3)^{-1/4} = 3^{-3/4} \)
- \( (9)^{1/6} = (3^2)^{1/6} = 3^{2/6} = 3^{1/3} \)
- \( (0.81)^{2/3} = (81^{-1})^{2/3} = (3^4)^{-2/3} = 3^{-8/3} \)

2. **Combine the terms:**
\[ \text{Numerator} = \frac{3^{1/3}}{10^{1/3}} \cdot 3^{-3/4} \cdot 3^{1/3} \cdot 3^{-8/3} \]
- Combine the powers of 3:
\[ 3^{1/3 - 3/4 + 1/3 - 8/3} = 3^{(4/12 - 9/12 + 4/12 - 32/12)} = 3^{-33/12} = 3^{-11/4} \]
- Thus, the numerator simplifies to:
\[ \frac{3^{-11/4}}{10^{1/3}} \]

### Step 2: Simplify the Denominator

1. **Calculate each term in the denominator:**
- \( (0.9)^{2/3} = (9/10)^{2/3} = \frac{9^{2/3}}{10^{2/3}} = \frac{(3^2)^{2/3}}{10^{2/3}} = \frac{3^{4/3}}{10^{2/3}} \)
- \( (3)^{-1/2} = 3^{-1/2} \)
- \( (1/3)^{-2} = 3^{2} \)
- \( (243)^{-1/4} = (3^5)^{-1/4} = 3^{-5/4} \)

2. **Combine the terms:**
\[ \text{Denominator} = \frac{3^{4/3}}{10^{2/3}} \cdot 3^{-1/2} \cdot 3^{2} \cdot 3^{-5/4} \]
- Combine the powers of 3:
\[ 3^{4/3 - 1/2 + 2 - 5/4} = 3^{(16/12 - 6/12 + 24/12 - 15/12)} = 3^{19/12} \]
- Thus, the denominator simplifies to:
\[ \frac{3^{19/12}}{10^{2/3}} \]

### Step 3: Combine Numerator and Denominator

Now, we can write the entire expression as:

\[ \frac{\frac{3^{-11/4}}{10^{1/3}}}{\frac{3^{19/12}}{10^{2/3}}} = \frac{3^{-11/4}}{10^{1/3}} \cdot \frac{10^{2/3}}{3^{19/12}} = \frac{3^{-11/4}}{3^{19/12}} \cdot 10^{2/3 - 1/3} \]

### Step 4: Simplify Further

1. **Combine the powers of 10:**
\[ 10^{2/3 - 1/3} = 10^{1/3} \]

2. **Combine the powers of 3:**
\[ 3^{-11/4 - 19/12} = 3^{(-33/12 - 19/12)} = 3^{-52/12} = 3^{-13/3} \]

Thus, the expression simplifies to:

\[ \frac{10^{1/3}}{3^{13/3}} = \frac{1}{3^{13/3}} \cdot 10^{1/3} \]

### Final Step: Evaluate the Expression

The final expression can be simplified to:

\[ \frac{10^{1/3}}{3^{13/3}} = \frac{(10/3^{13})^{1/3}}{1} \]

### Conclusion

The value of the expression is:

\[ \frac{1}{3} = 0.3 \]