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Radical Expressions and Equations

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

Infinitely Nested Radicals

         

What's the correct statement about \(x\)?

\[ x = \sqrt{x - \sqrt{x - \sqrt{x \ldots } }} \]

Hint: The expression marked in purple below is equivalent to \( x \). Use substitution!

\[ x = \sqrt{\sqrt{3\sqrt{\sqrt{3\sqrt{\sqrt{3} \ldots}}}}} \]

Which of these is a solution for \(x\)?

Hint: The expression marked in purple below is equivalent to \( x \). Use substitution!

\[ x = \sqrt{2\sqrt[3]{5\sqrt{2\sqrt[3]{5 \ldots}}}} \]

Which of these is a solution for \(x\)?

\[ x = \sqrt{2 \sqrt{x \sqrt{3 \sqrt{2 \sqrt{x \sqrt{3 \ldots } }}}}} \]

What's the nonzero solution for \( x \)?

\[ x = \sqrt{a \sqrt{a \sqrt{a \sqrt{a \ldots }}}} \]

If \( a \) is nonzero, which of these does \( x \) equal?

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