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Polynomial Arithmetic

From profit/cost models to the flight of a baseball to approximating the motion of a wave, polynomials are a mathematical way to represent values which are sums of powers of variables.

Level 3

         

\[\sqrt{\sqrt{\sqrt[3]{\color{blue}{64000}} + {\sqrt[3]{\color{blue}{64000} + \color{teal}{3(1640) + 1}}}}}\]

With the assistance of the algebraic identity \( (a+b)^3 = a^3 + b^3 + 3ab(a+b) \), evaluate the above expression.

If \(a\) and \(b\) are non-zero real numbers, simplify: \[\left(a + \frac1a\right)^2+\left(b + \frac1b\right)^2 + \left(ab + \frac1{ab}\right)^2 \\ - \left(a + \frac1a\right)\left(b + \frac1b\right)\left(ab + \frac1{ab}\right) \]

A lazy mathematician believes that

\[a^2 + b^2 = (a + b)^2.\]

If \(a\) and \(b\) are both integers chosen from the interval \(\left[-100, 100\right]\), then find the number of ordered pairs \((a, b)\) that satisfy the equation above.

\[\sqrt[3]{ 1+ \sqrt{x}} + \sqrt[3]{1 - \sqrt{x}} = \sqrt[3]{5}\]

One solution is of the form \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers.

Find \(a+b\).

\[ \begin{eqnarray} \large \color{blue}2^{\color{brown}x} - \color{blue}2^{\color{brown}y} & = & \large 1 \\ \\ \large \color{green}4^{\color{brown}x}- \color{green}4^{\color{brown}y} & = & \large { \frac 5 3} \\ \\ \\ \large {\color{brown}x} - {\color{brown}y} & = & \large \ \color{grey}? \end{eqnarray} \]

Details and Assumptions:

\(x\) and \(y\) are real numbers.

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