If `x = (3)/(2) ` rthen the value of `27 x ^(3) - 54 x ^(2) + 36 x - 11 `is

Options

A. `11 (3)/(8)`
B. `11 (5)/(8)`
C. `12 (3)/(8)`
D. `12 (5)/(8)`

Answer

To find the value of the expression \( 27x^3 - 54x^2 + 36x - 11 \) when \( x = \frac{3}{2} \), we will substitute \( x \) into the expression and simplify step by step.

### Step 1: Substitute \( x \) into the expression
We start with the expression:
\[ 27x^3 - 54x^2 + 36x - 11 \]
Substituting \( x = \frac{3}{2} \):
\[ 27\left(\frac{3}{2}\right)^3 - 54\left(\frac{3}{2}\right)^2 + 36\left(\frac{3}{2}\right) - 11 \]

### Step 2: Calculate \( x^3 \) and \( x^2 \)
Calculate \( \left(\frac{3}{2}\right)^3 \) and \( \left(\frac{3}{2}\right)^2 \):
\[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]
\[ \left(\frac{3}{2}\right)^3 = \frac{27}{8} \]

### Step 3: Substitute back into the expression
Now substitute these values back into the expression:
\[ 27\left(\frac{27}{8}\right) - 54\left(\frac{9}{4}\right) + 36\left(\frac{3}{2}\right) - 11 \]

### Step 4: Simplify each term
Now we simplify each term:
1. \( 27 \times \frac{27}{8} = \frac{729}{8} \)
2. \( 54 \times \frac{9}{4} = \frac{486}{4} = \frac{972}{8} \)
3. \( 36 \times \frac{3}{2} = 54 \)

Now, convert \( 54 \) to have a denominator of 8:
\[ 54 = \frac{432}{8} \]

### Step 5: Combine all terms
Now we can combine all the terms:
\[ \frac{729}{8} - \frac{972}{8} + \frac{432}{8} - \frac{88}{8} \]
Combine the numerators:
\[ 729 - 972 + 432 - 88 = 729 - 972 = -243 \]
\[ -243 + 432 = 189 \]
\[ 189 - 88 = 101 \]

### Step 6: Write the final result
Now we have:
\[ \frac{101}{8} \]

### Conclusion
Thus, the value of \( 27x^3 - 54x^2 + 36x - 11 \) when \( x = \frac{3}{2} \) is:
\[ \frac{101}{8} \]