Simplify : `sqrt(3sqrt(3sqrt(3sqrt(3sqrt(3)))))`

Options

A. `3^(15//16)`
B. `3^(33//32)`
C. `3^(21//32)`
D. `3^(31//32)`

Answer

To simplify the expression \( \sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}} \), we can follow these steps:

### Step 1: Understand the structure of the expression
The expression consists of nested square roots. We can denote the entire expression as \( x \):
\[ x = \sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}} \]

### Step 2: Rewrite the expression
Notice that we can express the nested square roots in terms of powers:
\[ x = \sqrt{3 \cdot x_1} \]
where \( x_1 = \sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}} \).

### Step 3: Continue breaking it down
We can continue this process. Let’s denote each nested square root:
- \( x_1 = \sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}} \)
- \( x_2 = \sqrt{3\sqrt{3\sqrt{3}}} \)
- \( x_3 = \sqrt{3\sqrt{3}} \)
- \( x_4 = \sqrt{3} \)

### Step 4: Establish a recursive relationship
We can express this recursively:
\[ x_n = \sqrt{3 \cdot x_{n-1}} \]
where \( x_0 = 3 \) (the innermost value).

### Step 5: Generalize the expression
From the recursive relationship, we can derive:
\[ x_n = 3^{1/2} \cdot x_{n-1}^{1/2} \]
This means:
\[ x_n = 3^{1/2^n} \cdot 3^{1/2^{n-1}} \cdots 3^{1/2^1} \cdot 3^{1/2^0} \]
This can be simplified to:
\[ x_n = 3^{(1/2 + 1/4 + 1/8 + \ldots + 1/2^n)} \]

### Step 6: Sum the series
The series \( 1/2 + 1/4 + 1/8 + \ldots + 1/2^n \) is a geometric series with the first term \( a = 1/2 \) and the common ratio \( r = 1/2 \). The sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Thus:
\[ S_n = \frac{1/2(1 - (1/2)^n)}{1/2} = 1 - \frac{1}{2^n} \]

### Step 7: Substitute back into the expression
Now substituting back, we have:
\[ x_n = 3^{(1 - \frac{1}{2^n})} \]

### Step 8: Evaluate for \( n = 5 \)
For our case, since we have 5 nested square roots:
\[ x_5 = 3^{(1 - \frac{1}{2^5})} = 3^{(1 - \frac{1}{32})} = 3^{\frac{31}{32}} \]

### Final Result
Thus, the simplified form of the expression \( \sqrt{3\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}}} \) is:
\[ \boxed{3^{\frac{31}{32}}} \]