The value of expression `((0.6)^(0)-(0.1)^(-1))/(((3)/(2^(3)))^(-1).((3)/(2))^(3)+(-(1)/(3))^(-1))` is

Options

A. `-3//2`
B. `2//3`
C. `3//2`
D. `9//4`

Answer

To solve the expression \(\frac{(0.6)^{0} - (0.1)^{-1}}{((\frac{3}{2^{3}})^{-1} \cdot (\frac{3}{2})^{3}) + (-(\frac{1}{3}))^{-1}}\), we will break it down step by step.

### Step 1: Simplify the numerator
The numerator is \((0.6)^{0} - (0.1)^{-1}\).

- We know that any number raised to the power of 0 is 1:
\[ (0.6)^{0} = 1 \]
- For \((0.1)^{-1}\), we can rewrite it as the reciprocal:
\[ (0.1)^{-1} = \frac{1}{0.1} = 10 \]
- Now substituting these values back into the numerator:
\[ 1 - 10 = -9 \]

### Step 2: Simplify the denominator
The denominator is \(((\frac{3}{2^{3}})^{-1} \cdot (\frac{3}{2})^{3}) + (-(\frac{1}{3}))^{-1}\).

- First, simplify \((\frac{3}{2^{3}})^{-1}\):
\[ (\frac{3}{2^{3}})^{-1} = \frac{2^{3}}{3} \]
- Next, simplify \((\frac{3}{2})^{3}\):
\[ (\frac{3}{2})^{3} = \frac{3^{3}}{2^{3}} = \frac{27}{8} \]
- Now, multiply these two results:
\[ \frac{2^{3}}{3} \cdot \frac{27}{8} = \frac{8}{3} \cdot \frac{27}{8} = \frac{27}{3} = 9 \]

- Now simplify \((-(\frac{1}{3}))^{-1}\):
\[ (-(\frac{1}{3}))^{-1} = -3 \]

- Now substitute these values back into the denominator:
\[ 9 + (-3) = 9 - 3 = 6 \]

### Step 3: Combine the results
Now we have:
\[ \text{Numerator} = -9 \]
\[ \text{Denominator} = 6 \]
So the expression simplifies to:
\[ \frac{-9}{6} = -\frac{3}{2} \]

### Final Answer
Thus, the value of the expression is:
\[ -\frac{3}{2} \]