Answer :

To simplify the expression [tex]\frac{186 \times 186 + 14 \times 14}{186 \times 186 - 186 \times 14 + 14 \times 14}[/tex] using an identity, we can observe the structure of the expression.

The numerator [tex]186 \times 186 + 14 \times 14[/tex] can be written as [tex]186^2 + 14^2[/tex] and the denominator [tex]186 \times 186 - 186 \times 14 + 14 \times 14[/tex] can be written as [tex]186^2 - 186 \times 14 + 14^2[/tex].

Notice that both these expressions resemble parts of the identity [tex]a^2 + b^2[/tex] in the numerator and [tex]a^2 - ab + b^2[/tex] in the denominator. The denominator is actually a perfect example of the quadratic expression [tex](a - b)^2[/tex].

Let's use these observations to simplify:

  1. Numerator:
    [tex]186^2 + 14^2[/tex]

  2. Denominator:
    [tex]186^2 - 186 \times 14 + 14^2 = (186 - 14)^2[/tex]

  3. Simplifying using identity:
    The numerator is [tex]a^2 + b^2[/tex] and the denominator simplifies to [tex](a-b)^2[/tex] which is:
    [tex](186 - 14)^2 = 172^2[/tex]

  4. Substitute back into the expression, you get:
    [tex]\frac{186^2 + 14^2}{172^2}[/tex]

The expression can't be simplified further without more specific requirements. This form is the simplest representation using identities known to match the observed structure.

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Rewritten by : Barada