Simplify the following expression: \(\dfrac{(25)^4+1250+1-(25)^2}{(25)^2+26}\)
601
Let \(a = 25\) so the expression becomes easier to handle. Note that \(1250 = 2\times 625 = 2\times 25^2 = 2a^2\).
Rewrite the numerator: \((25)^4 + 1250 + 1 - (25)^2 = a^4 + 2a^2 + 1 - a^2\).
Combine the \(a^2\) terms: \(a^4 + 2a^2 - a^2 + 1 = a^4 + a^2 + 1\).
Apply the standard factorisation \(a^4 + a^2 + 1 = (a^2 + a + 1)(a^2 - a + 1)\).
Substitute \(a = 25\): \((625 + 25 + 1)(625 - 25 + 1) = 651\times 601\).
The denominator is \(25^2 + 26 = 625 + 26 = 651\).
So the value \(= \frac{651\times 601}{651} = 601\), since the factor 651 cancels.
Key concept: the identity \(a^4 + a^2 + 1 = (a^2+a+1)(a^2-a+1)\) lets a large fraction cancel neatly.
Hence the simplified value is 601.
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