When `x = (1)/(9) and y = (-3)/(4),` then the value of the expression
`81 x ^(2) + 16 y ^(2)- 72 xy` is

Options

A. 25
B. `(9)/(16)`
C. 16
D. `(9)/(4)`

Answer

To find the value of the expression \( 81x^2 + 16y^2 - 72xy \) when \( x = \frac{1}{9} \) and \( y = -\frac{3}{4} \), we will substitute the values of \( x \) and \( y \) into the expression and simplify step by step.

### Step 1: Substitute the values of \( x \) and \( y \)

We start with the expression:
\[ 81x^2 + 16y^2 - 72xy \]

Substituting \( x = \frac{1}{9} \) and \( y = -\frac{3}{4} \):
\[ 81\left(\frac{1}{9}\right)^2 + 16\left(-\frac{3}{4}\right)^2 - 72\left(\frac{1}{9}\right)\left(-\frac{3}{4}\right) \]

### Step 2: Calculate \( x^2 \) and \( y^2 \)

Calculating \( x^2 \):
\[ x^2 = \left(\frac{1}{9}\right)^2 = \frac{1}{81} \]

Calculating \( y^2 \):
\[ y^2 = \left(-\frac{3}{4}\right)^2 = \frac{9}{16} \]

### Step 3: Substitute \( x^2 \) and \( y^2 \) back into the expression

Now substitute \( x^2 \) and \( y^2 \) back into the expression:
\[ 81 \cdot \frac{1}{81} + 16 \cdot \frac{9}{16} - 72 \cdot \frac{1}{9} \cdot \left(-\frac{3}{4}\right) \]

### Step 4: Simplify each term

Calculating the first term:
\[ 81 \cdot \frac{1}{81} = 1 \]

Calculating the second term:
\[ 16 \cdot \frac{9}{16} = 9 \]

Calculating the third term:
\[ 72 \cdot \frac{1}{9} \cdot \left(-\frac{3}{4}\right) = -8 \cdot (-3) = 6 \]

### Step 5: Combine all the terms

Now combine all the terms:
\[ 1 + 9 + 6 = 16 \]

### Final Answer

Thus, the value of the expression \( 81x^2 + 16y^2 - 72xy \) is:
\[ \boxed{16} \]