Algebra
# Polynomial Arithmetic

$\sqrt{\sqrt{\sqrt[3]{\color{#3D99F6}{64000}} + {\sqrt[3]{\color{#3D99F6}{64000} + \color{teal}{3(1640) + 1}}}}}$

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

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 of the above equation is of the form $\frac{a}{b},$ where $a$ and $b$ are coprime positive integers.

Find $a+b$.

$\begin{aligned} \large \color{#3D99F6}2^{\color{#624F41}x} - \color{#3D99F6}2^{\color{#624F41}y} & = & \large 1 \\ \\ \large \color{#20A900}4^{\color{#624F41}x}- \color{#20A900}4^{\color{#624F41}y} & = & \large { \frac 5 3} \\ \\ \\ \large {\color{#624F41}x} - {\color{#624F41}y} & = & \large \, ? \end{aligned}$

**Details and Assumptions:**

- $x$ and $y$ are real numbers.