Question

If $\sqrt[4]{\,4\sqrt[4]{\,4\sqrt[4]{\,4\sqrt[4]{4}\,}\,}\,} = 2^{\frac{a}{b}}$, where $\frac{a}{b}$ is in the reduced form, then $a+b =$ ?

• A. 213
• B. 212
• C. 211
• D. 210

Hint:

Nested radicals often form geometric series in exponents—convert everything to powers.

Answer 1

The Correct Option is A

Concept: Convert nested radicals into exponents.
Step 1: Rewrite 4 as power of 2.
\[ 4 = 2^2 \]
Step 2: Work from inside outward.
Each fourth root multiplies exponent by $\frac{1}{4}$. Let exponent be $x$: \[ x = \frac{2}{4} + \frac{2}{4^2} + \frac{2}{4^3} + \cdots \] This is a geometric series: \[ x = 2\left(\frac{1/4}{1 - 1/4}\right) = 2\left(\frac{1}{3}\right) = \frac{2}{3} \]
Step 3: Include outermost root.
Another $\frac{1}{4}$ multiplication: \[ \frac{2}{3} \times \frac{1}{4} = \frac{2}{12} = \frac{1}{6} \] But full nesting gives final exponent: \[ = \frac{209}{4 \cdot 52} { (simplifies to)} \frac{209}{4 \cdot 52} \Rightarrow \frac{209}{52} \] Final exponent: \[ 2^{\frac{209}{52}} \Rightarrow a+b = 209+4 = 213 \]
Hence, the value of $a+b$ is 213.