Solve the followings :
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

\[ \frac{(0.6)^0 - (0.1)^{-1}}{(3/(2^3))^{-1} \cdot (3/2)^{3} + (-1/3)^{-1}} \]

we will break it down step by step.

### Step 1: Simplify the numerator

The numerator is

\[ (0.6)^0 - (0.1)^{-1} \]

Using the property that any number raised to the power of 0 is 1, we have:

\[ (0.6)^0 = 1 \]

Now, for \((0.1)^{-1}\), we can rewrite it as:

\[ (0.1)^{-1} = \frac{1}{(0.1)^1} = \frac{1}{0.1} = 10 \]

So, the numerator becomes:

\[ 1 - 10 = -9 \]

### Step 2: Simplify the denominator

The denominator is

\[ (3/(2^3))^{-1} \cdot (3/2)^{3} + (-1/3)^{-1} \]

First, simplify \((3/(2^3))^{-1}\):

\[ (3/(2^3))^{-1} = \frac{2^3}{3} = \frac{8}{3} \]

Next, simplify \((3/2)^{3}\):

\[ (3/2)^{3} = \frac{3^3}{2^3} = \frac{27}{8} \]

Now, multiply these two results:

\[ \frac{8}{3} \cdot \frac{27}{8} = \frac{27}{3} = 9 \]

Next, simplify \((-1/3)^{-1}\):

\[ (-1/3)^{-1} = -3 \]

Now, combine these two results in the denominator:

\[ 9 + (-3) = 9 - 3 = 6 \]

### Step 3: Combine the results

Now we can substitute the simplified numerator and denominator back into the expression:

\[ \frac{-9}{6} \]

This simplifies to:

\[ -\frac{3}{2} \]

### Final Answer

Thus, the value of the expression is

\[ -\frac{3}{2} \]