Simplify : `(5.32xx56+5.32xx44)/((7.66)^(2)-(2.34)^(2))`

Options

A. A) `7.2`
B. B) `8.5`
C. C) `10`
D. D) `12`

Answer

To simplify the expression \((5.32 \times 56 + 5.32 \times 44) / ((7.66)^2 - (2.34)^2)\), we can follow these steps:

### Step 1: Factor out the common term in the numerator
The numerator \(5.32 \times 56 + 5.32 \times 44\) can be simplified by factoring out \(5.32\):
\[ 5.32 \times (56 + 44) \]

### Step 2: Simplify the sum in the parentheses
Now, calculate \(56 + 44\):
\[ 56 + 44 = 100 \]
So, the numerator becomes:
\[ 5.32 \times 100 \]

### Step 3: Simplify the denominator using the difference of squares
The denominator \((7.66)^2 - (2.34)^2\) can be simplified using the difference of squares formula \(A^2 - B^2 = (A + B)(A - B)\):
Let \(A = 7.66\) and \(B = 2.34\):
\[ (7.66 + 2.34)(7.66 - 2.34) \]

### Step 4: Calculate \(A + B\) and \(A - B\)
Now, calculate \(7.66 + 2.34\) and \(7.66 - 2.34\):
\[ 7.66 + 2.34 = 10 \]
\[ 7.66 - 2.34 = 5.32 \]
Thus, the denominator becomes:
\[ 10 \times 5.32 \]

### Step 5: Substitute back into the expression
Now we can substitute the simplified numerator and denominator back into the expression:
\[ \frac{5.32 \times 100}{10 \times 5.32} \]

### Step 6: Cancel the common terms
We can cancel \(5.32\) from the numerator and the denominator:
\[ \frac{100}{10} \]

### Step 7: Simplify the fraction
Now, simplify \(\frac{100}{10}\):
\[ 100 \div 10 = 10 \]

### Final Answer
The simplified expression is:
\[ 10 \]

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