Find the value of the expression `(81 x^(2) + 16 y^(2) - 72 xy)` when `x = (2)/(3)` and `y = (3)/(4)`

Answer

To find the value of the expression \( 81x^2 + 16y^2 - 72xy \) when \( x = \frac{2}{3} \) and \( y = \frac{3}{4} \), we will follow these steps:

### Step 1: Substitute the values of \( x \) and \( y \) into the expression.
We have:
\[ x = \frac{2}{3} \quad \text{and} \quad y = \frac{3}{4} \]
Substituting these values into the expression:
\[ 81\left(\frac{2}{3}\right)^2 + 16\left(\frac{3}{4}\right)^2 - 72\left(\frac{2}{3}\right)\left(\frac{3}{4}\right) \]

### Step 2: Calculate \( x^2 \) and \( y^2 \).
Calculating \( x^2 \):
\[ \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]
Calculating \( y^2 \):
\[ \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

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

### Step 4: Simplify each term.
Calculating the first term:
\[ 81 \cdot \frac{4}{9} = 9 \cdot 4 = 36 \]
Calculating the second term:
\[ 16 \cdot \frac{9}{16} = 9 \]
Calculating the third term:
\[ \frac{2}{3} \cdot \frac{3}{4} = \frac{2 \cdot 3}{3 \cdot 4} = \frac{2}{4} = \frac{1}{2} \]
So,
\[ 72 \cdot \frac{1}{2} = 36 \]

### Step 5: Combine all the terms.
Now we can combine the results:
\[ 36 + 9 - 36 \]

### Step 6: Perform the final calculation.
Calculating:
\[ 36 + 9 = 45 \]
Then,
\[ 45 - 36 = 9 \]

### Final Answer:
Thus, the value of the expression is \( 9 \).

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