If `a =2, b =-3` then the value of `27 a ^(3) - 54 a ^(2) + 36 b ^(2) - 8 b ^(3)` is

Options

A. 1562
B. 1616
C. 1676
D. 1728

Answer

To find the value of the expression \( 27a^3 - 54a^2 + 36b^2 - 8b^3 \) given \( a = 2 \) and \( b = -3 \), we will substitute the values of \( a \) and \( b \) into the expression and simplify step by step.

### Step 1: Write down the expression
The expression we need to evaluate is:
\[ 27a^3 - 54a^2 + 36b^2 - 8b^3 \]

### Step 2: Substitute the values of \( a \) and \( b \)
Substituting \( a = 2 \) and \( b = -3 \):
\[ 27(2)^3 - 54(2)^2 + 36(-3)^2 - 8(-3)^3 \]

### Step 3: Calculate \( a^3 \) and \( a^2 \)
Calculate \( 2^3 \) and \( 2^2 \):
\[ 2^3 = 8 \quad \text{and} \quad 2^2 = 4 \]

### Step 4: Calculate \( b^2 \) and \( b^3 \)
Calculate \( (-3)^2 \) and \( (-3)^3 \):
\[ (-3)^2 = 9 \quad \text{and} \quad (-3)^3 = -27 \]

### Step 5: Substitute back into the expression
Now substitute these values back into the expression:
\[ 27(8) - 54(4) + 36(9) - 8(-27) \]

### Step 6: Calculate each term
Now calculate each term:
- \( 27(8) = 216 \)
- \( -54(4) = -216 \)
- \( 36(9) = 324 \)
- \( -8(-27) = 216 \)

### Step 7: Combine the results
Now combine all these results:
\[ 216 - 216 + 324 + 216 \]

### Step 8: Simplify
Now simplify:
\[ 0 + 324 + 216 = 540 \]

### Final Result
Thus, the value of the expression is:
\[ \boxed{540} \]