Algebra

# Radical Expressions and Equations: Level 3 Challenges

$\LARGE \sqrt{3} - \sqrt{5}$

Which one of these choices is equivalent to the expression above?

$\large \frac{4}{\big(\sqrt{5} + 1\big)\big(\sqrt[4]{5} + 1\big)\big(\sqrt[8]{5} + 1\big)\big(\sqrt[16]{5} + 1\big)}$

If the above expression is equal to $$x$$, find the value of $$(x+1)^{64}$$.

$\sqrt{3+\sqrt{3}+\sqrt{2+\sqrt{3}+\sqrt{7+\sqrt{48}}}} = a + \sqrt{b}$

If $$a$$ and $$b$$ are positive integers, find $$a + b$$.

$\large \sqrt{4+\sqrt{4-\sqrt{ 4+\sqrt{4-\sqrt{ 4+\sqrt{4-\dots}}}}}}$

What is the closed form of the above expression?

The equation $$x + \sqrt{x-2} = 4$$ has $$\text{__________}.$$

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