There are three key exponent properties needed to understand this. When you multiply terms with the same base, you add their exponents. 2. When raise a term to a certain power, you multiply the ...

This is a special case of the familiar identity a^2+2ab+b^2=(a+b)^2, with a=x^2 and b=\frac{1}{x^2}. That makes the "middle" term 2ab equal to 2. The easiest way to see things is ...

Let \{s_m:m\in\mathbb{N}\} be enumeration of \mathbb{Q}\setminus\{-32^{-m}:m\in\mathbb{N}\}. Define enumeration of \mathbb{Q}: r_k= \begin{cases} -32^{-m}&\quad\text{ if }\quad k=2m\\ s_m&\quad\text{ if }\quad k=2m-1 \end{cases} ...

If the radius of the cone is x\leq R then the distance of the base of the cone from the center of the sphere is given by \sqrt{R^2-x^2} and the height of the cone is R+\sqrt{R^2-x^2}. If we set ...