When `a = (4)/(3),` the value of `27 a ^(3)- 108 a ^(2) + 141 a - 31 ` is

Options

A. A)261
B. B)`-253`
C. C)`-245`
D. D)`29`

Answer

To find the value of the expression \( 27a^3 - 108a^2 + 141a - 31 \) when \( a = \frac{4}{3} \), we will substitute \( a \) into the expression and simplify step by step.

### Step 1: Substitute \( a \) into the expression
We start with the expression:
\[ 27a^3 - 108a^2 + 141a - 31 \]
Substituting \( a = \frac{4}{3} \):
\[ 27\left(\frac{4}{3}\right)^3 - 108\left(\frac{4}{3}\right)^2 + 141\left(\frac{4}{3}\right) - 31 \]

### Step 2: Calculate \( a^3 \) and \( a^2 \)
First, we calculate \( a^3 \) and \( a^2 \):
\[ a^3 = \left(\frac{4}{3}\right)^3 = \frac{4^3}{3^3} = \frac{64}{27} \]
\[ a^2 = \left(\frac{4}{3}\right)^2 = \frac{4^2}{3^2} = \frac{16}{9} \]

### Step 3: Substitute \( a^3 \) and \( a^2 \) back into the expression
Now substitute these values back:
\[ 27\left(\frac{64}{27}\right) - 108\left(\frac{16}{9}\right) + 141\left(\frac{4}{3}\right) - 31 \]

### Step 4: Simplify each term
1. The first term:
\[ 27 \cdot \frac{64}{27} = 64 \]
2. The second term:
\[ 108 \cdot \frac{16}{9} = 12 \cdot 16 = 192 \]
3. The third term:
\[ 141 \cdot \frac{4}{3} = \frac{564}{3} = 188 \]
4. The last term remains as is:
\[ -31 \]

### Step 5: Combine all the terms
Now we combine all the simplified terms:
\[ 64 - 192 + 188 - 31 \]

### Step 6: Perform the arithmetic
1. Combine \( 64 - 192 \):
\[ 64 - 192 = -128 \]
2. Add \( 188 \):
\[ -128 + 188 = 60 \]
3. Subtract \( 31 \):
\[ 60 - 31 = 29 \]

### Final Answer
The value of the expression when \( a = \frac{4}{3} \) is:
\[ \boxed{29} \]