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A factored polynomial reveals its roots, a key concept in understanding the behavior of these expressions.

\[ \large \sqrt a + b = 7 \\ \large \sqrt b + a = 11 \]

If \(a\) and \(b\) are real numbers that satisfy the equation above, what is the value of \(a\) and \(b\) respectively?

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Given that \(6y^2=2014\), find the value of \[\frac{9(y^4+6y^3+9y^2)}{y^2+6y+9}. \]

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\[\large a^3 + b^3 = 2593080, \ \ a + b = 210, \ \ \ \ \ ab = \ ? \]

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Given that \(a+b=1\) and \(a^2+b^2=2\), what is the value of \(a^7+b^7\)?

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If \( x^2 - 5x - 1 = 0 \), then find the value of \( x^2 + \frac{1}{x^2}.\)

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