The fourth power of √1+√1+√1 is:

1970 AMC 12 Problem 1

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The fourth power of

\sqrt{1+\sqrt{1+\sqrt1}}

is:

\sqrt2+\sqrt3

\tfrac12(7+3\sqrt5)

1+2\sqrt3

3

3+2\sqrt2

Simplify from the innermost radical outward

After the innermost radical becomes

1,

square

\sqrt{1+\sqrt2}

twice

The innermost radical is

\sqrt1=1,

so the given number is

\sqrt{1+\sqrt2}.

Its fourth power is

\begin{aligned} (1+\sqrt2)^2&=1+2\sqrt2+2\\ &=3+2\sqrt2. \end{aligned}

Therefore, the correct answer is E.