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# Radical Expressions and Equations

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

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\[\LARGE \sqrt{3} - \sqrt{5} \]

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

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Paul Ryan Longhas**

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

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

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Matthew Kendall**

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

If \(a\) and \(b\) are positive integers, find \( a + b \).

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Rohit Udaiwal**

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\[ \large \sqrt{4+\sqrt{4-\sqrt{ 4+\sqrt{4-\sqrt{ 4+\sqrt{4-\dots}}}}}}\]

What is the closed form of the above expression?

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Brilliant Member**

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

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Soumo Mukherjee**

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