`((6.4)^(2)-(5.4)^(2))/((8.9)^(2)+8.9xx2.2+(1.1)^(2))=`

Options

A. `0.118`
B. `0.92`
C. `1.5`
D. `0.61`

Answer

To solve the expression \(\frac{(6.4)^2 - (5.4)^2}{(8.9)^2 + 8.9 \times 2.2 + (1.1)^2}\), we will use algebraic identities.

### Step-by-Step Solution:

1. **Identify the Numerator**:
The numerator is \( (6.4)^2 - (5.4)^2 \). We can apply the difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
Here, \( a = 6.4 \) and \( b = 5.4 \).

**Calculation**:
\[ 6.4 - 5.4 = 1.0 \]
\[ 6.4 + 5.4 = 11.8 \]
Therefore,
\[ (6.4)^2 - (5.4)^2 = (1.0)(11.8) = 11.8 \]

**Hint**: Use the difference of squares formula to simplify expressions of the form \( a^2 - b^2 \).

2. **Identify the Denominator**:
The denominator is \( (8.9)^2 + 8.9 \times 2.2 + (1.1)^2 \). We can recognize this as a perfect square trinomial:
\[ a^2 + 2ab + b^2 = (a + b)^2 \]
Here, \( a = 8.9 \) and \( b = 1.1 \).

**Calculation**:
\[ 8.9 + 1.1 = 10.0 \]
Therefore,
\[ (8.9)^2 + 2 \times 8.9 \times 1.1 + (1.1)^2 = (10.0)^2 = 100 \]

**Hint**: Recognize patterns in the denominator that match the perfect square trinomial formula.

3. **Combine the Results**:
Now we can substitute the simplified numerator and denominator back into the expression:
\[ \frac{11.8}{100} \]

4. **Final Calculation**:
Simplifying this gives:
\[ 0.118 \]

### Final Answer:
\[ \frac{(6.4)^2 - (5.4)^2}{(8.9)^2 + 8.9 \times 2.2 + (1.1)^2} = 0.118 \]