$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}=\frac{a+\sqrt{b}}{c}$

Find the values for $a, b$ and $c$ and write your answer in the form $\frac{a+\sqrt{b}}{c}$.

$\sqrt[3]{3^{3^3}}={\large ?}$

*Give your answer in the form $3^a$ where $a$ is an integer.

Calculating square roots without a calculator
Look at the following article on how to calculate $\sqrt{7}$:
$$
\require{enclose}
\begin{array}{l}
\phantom{2}\;2.\;\;6\;\;\,4\;\;\,5\;\;\,\ldots \\[-3pt]
\sqrt{7.00’00’00’00\ldots} \\[-3pt]
\phantom{2}\;\underline{4}\\[-3pt]
\phantom{2}\;300\\[-3pt]
\phantom{2}\;\underline{276}\\[-3pt]
\phantom{2}\phantom{2}\;2400\\[-3pt]
\phantom{2}\phantom{2}\;\underline{2096}\\[-3pt]
\phantom{2}\phantom{2}\phantom{2}\;30400\\[-3pt]
\phantom{2}\phantom{2}\phantom{2}\;\underline{26425}\\[-3pt]
\phantom{2}\phantom{2}\phantom{2}\phantom{2}\;397500\ldots\\
\end{array}
$$
Use this method to calculate the following to 3 decimal places:
a) $\sqrt{2}$
b) $\sqrt{11}$
c) $\sqrt{999}$

Find all of the positive integer values of $n$ that makes $\sqrt{n^2+55}$ an integer.
*Separate the values with commas

Write $\sqrt{5-2\sqrt{6}}$ in the form $\sqrt{a}-\sqrt{b}$ where $a$ and $b$ are integers.

Without listing all the factors, can you find how many factors each number has?
a) 2401
b) 168
c) 1764

Simplify.
$\left(x^\frac{1}{a-b}\right)^\frac{1}{a-c} \left(x^\frac{1}{b-c}\right)^\frac{1}{b-a} \left(x^\frac{1}{c-a}\right)^\frac{1}{c-b}$

If $x^\frac{1}{2}+x^{-\frac{1}{2}}=a$, write $x^2+x^{-2}$ in terms of $a$.

Find the solution that is not $x=0$ or $x=1$ for the following:
$$\left(\sqrt[4]{x}\right)^{4x^4}=\left(x^4\right)^{4\sqrt[4]{x}}$$
Give your answer as an exponent.

Solve for $x$.
$4^{\sqrt{x^2-2}+x}-5\cdot2^{x-1+\sqrt{x^2-2}}=6$