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

Options

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

Answer

To solve the expression \(81x^2 + 16y^2 - 72xy\) when \(x = \frac{1}{9}\) and \(y = -\frac{3}{4}\), we can follow these steps:

### Step 1: Substitute the values of \(x\) and \(y\)
We start by substituting \(x\) and \(y\) into the expression:
\[ 81\left(\frac{1}{9}\right)^2 + 16\left(-\frac{3}{4}\right)^2 - 72\left(\frac{1}{9}\right)\left(-\frac{3}{4}\right) \]

### Step 2: Calculate \(81x^2\)
First, calculate \(81x^2\):
\[ 81\left(\frac{1}{9}\right)^2 = 81 \cdot \frac{1}{81} = 1 \]

### Step 3: Calculate \(16y^2\)
Next, calculate \(16y^2\):
\[ 16\left(-\frac{3}{4}\right)^2 = 16 \cdot \frac{9}{16} = 9 \]

### Step 4: Calculate \(-72xy\)
Now, calculate \(-72xy\):
\[ -72\left(\frac{1}{9}\right)\left(-\frac{3}{4}\right) = 72 \cdot \frac{1}{9} \cdot \frac{3}{4} = 8 \cdot 3 = 24 \]

### Step 5: Combine the results
Now, we combine all the results:
\[ 1 + 9 + 24 = 34 \]

### Final Answer
Thus, the value of the expression \(81x^2 + 16y^2 - 72xy\) is:
\[ \boxed{34} \]