Evaluate:
`((7)/(8)x + (4)/(5)y)^(2)`

Answer

To evaluate the expression \(\left(\frac{7}{8}x + \frac{4}{5}y\right)^{2}\), we will use the formula for the square of a binomial:

\[ (a + b)^{2} = a^{2} + b^{2} + 2ab \]

**Step 1: Identify \(a\) and \(b\)**
In our case, let:
- \(a = \frac{7}{8}x\)
- \(b = \frac{4}{5}y\)

**Step 2: Calculate \(a^{2}\)**
Now we compute \(a^{2}\):
\[ a^{2} = \left(\frac{7}{8}x\right)^{2} = \frac{7^{2}}{8^{2}}x^{2} = \frac{49}{64}x^{2} \]

**Step 3: Calculate \(b^{2}\)**
Next, we calculate \(b^{2}\):
\[ b^{2} = \left(\frac{4}{5}y\right)^{2} = \frac{4^{2}}{5^{2}}y^{2} = \frac{16}{25}y^{2} \]

**Step 4: Calculate \(2ab\)**
Now we calculate \(2ab\):
\[ 2ab = 2 \cdot \left(\frac{7}{8}x\right) \cdot \left(\frac{4}{5}y\right) = 2 \cdot \frac{7 \cdot 4}{8 \cdot 5}xy = \frac{56}{40}xy = \frac{7}{5}xy \]

**Step 5: Combine all parts**
Now we can combine all the parts together:
\[ \left(\frac{7}{8}x + \frac{4}{5}y\right)^{2} = a^{2} + b^{2} + 2ab = \frac{49}{64}x^{2} + \frac{16}{25}y^{2} + \frac{7}{5}xy \]

Thus, the final answer is:
\[ \frac{49}{64}x^{2} + \frac{16}{25}y^{2} + \frac{7}{5}xy \]

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