\(\dfrac{(3.57)^2 + (4.54)^2 + 32.4156}{(4.54)^2 - (3.57)^2} = ?\)
8.36
Let \(a = 3.57\) and \(b = 4.54\) to simplify the expression.
Notice that \(32.4156 = 2 \times 3.57 \times 4.54 = 2ab\), so the numerator is \(a^2 + b^2 + 2ab\).
By the identity \(a^2 + b^2 + 2ab = (a + b)^2\), the numerator \(= (3.57 + 4.54)^2 = (8.11)^2\).
The denominator is a difference of squares: \(b^2 - a^2 = (b + a)(b - a) = (8.11)(0.97)\).
So the expression \(= \dfrac{(8.11)^2}{(8.11)(0.97)} = \dfrac{8.11}{0.97}\).
Dividing: \(\dfrac{8.11}{0.97} \approx 8.36\).
The value of the expression is 8.36.
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