Find the value of the expression `(36x^(2) + 25y^(2) - 60 xy)`, when `x = (2)/(3)` and `y = (1)/(5)`

Answer

To find the value of the expression \(36x^2 + 25y^2 - 60xy\) when \(x = \frac{2}{3}\) and \(y = \frac{1}{5}\), we will follow these steps:

### Step 1: Substitute the values of \(x\) and \(y\) into the expression.
We have:
\[ x = \frac{2}{3}, \quad y = \frac{1}{5} \]
Substituting these values into the expression:
\[ 36\left(\frac{2}{3}\right)^2 + 25\left(\frac{1}{5}\right)^2 - 60\left(\frac{2}{3}\right)\left(\frac{1}{5}\right) \]

### Step 2: Calculate \(x^2\) and \(y^2\).
Calculating \(x^2\):
\[ \left(\frac{2}{3}\right)^2 = \frac{4}{9} \]
Calculating \(y^2\):
\[ \left(\frac{1}{5}\right)^2 = \frac{1}{25} \]

### Step 3: Substitute \(x^2\) and \(y^2\) back into the expression.
Now substituting these values back into the expression:
\[ 36 \cdot \frac{4}{9} + 25 \cdot \frac{1}{25} - 60 \cdot \left(\frac{2}{3}\right) \cdot \left(\frac{1}{5}\right) \]

### Step 4: Simplify each term.
Calculating the first term:
\[ 36 \cdot \frac{4}{9} = \frac{144}{9} = 16 \]
Calculating the second term:
\[ 25 \cdot \frac{1}{25} = 1 \]
Calculating the third term:
\[ 60 \cdot \left(\frac{2}{3}\right) \cdot \left(\frac{1}{5}\right) = 60 \cdot \frac{2}{15} = \frac{120}{15} = 8 \]

### Step 5: Combine all the terms.
Now we combine all the calculated terms:
\[ 16 + 1 - 8 \]

### Step 6: Perform the final calculation.
Calculating:
\[ 16 + 1 = 17 \]
Then:
\[ 17 - 8 = 9 \]

### Final Answer:
The value of the expression \(36x^2 + 25y^2 - 60xy\) when \(x = \frac{2}{3}\) and \(y = \frac{1}{5}\) is \(9\).

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