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Radical Expressions and Equations

Combine your skills of working with exponents and manipulating expressions for radical glory.

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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