Find the values of `sqrt(6sqrt(6sqrt(6sqrt(6sqrt(6)))))`.

Options

A. 6
B. `6^(31//32)`
C. `6^(31//34)`
D. 3

Answer

To solve the expression \( \sqrt{6\sqrt{6\sqrt{6\sqrt{6\sqrt{6}}}}} \), we will simplify it step by step.

### Step 1: Rewrite the Expression
We start with the expression:
\[ \sqrt{6\sqrt{6\sqrt{6\sqrt{6\sqrt{6}}}}} \]
This can be rewritten as:
\[ \sqrt{6 \cdot \sqrt{6 \cdot \sqrt{6 \cdot \sqrt{6 \cdot \sqrt{6}}}}} \]

### Step 2: Simplify the Innermost Square Root
Let’s denote \( x = \sqrt{6\sqrt{6\sqrt{6\sqrt{6}}}} \). Thus, we have:
\[ \sqrt{6x} \]

### Step 3: Express \( x \) in Terms of Powers of 6
Now, we will simplify \( x \):
\[ x = \sqrt{6\sqrt{6\sqrt{6\sqrt{6}}}} = \sqrt{6 \cdot \sqrt{6 \cdot \sqrt{6 \cdot \sqrt{6}}}} \]
Continuing this process, we can express \( x \) in terms of powers of 6.

### Step 4: Recognize the Pattern
Notice that each square root can be expressed as a power of 6:
\[ \sqrt{6} = 6^{1/2} \]
\[ \sqrt{6\sqrt{6}} = \sqrt{6 \cdot 6^{1/2}} = \sqrt{6^{3/2}} = 6^{3/4} \]
\[ \sqrt{6\sqrt{6\sqrt{6}}} = \sqrt{6 \cdot 6^{3/4}} = \sqrt{6^{7/4}} = 6^{7/8} \]
\[ \sqrt{6\sqrt{6\sqrt{6\sqrt{6}}}} = \sqrt{6 \cdot 6^{7/8}} = \sqrt{6^{15/8}} = 6^{15/16} \]

### Step 5: Substitute Back into the Original Expression
Now substituting back into our expression:
\[ \sqrt{6 \cdot 6^{15/16}} = \sqrt{6^{1 + 15/16}} = \sqrt{6^{31/16}} \]

### Step 6: Final Simplification
Now we simplify:
\[ \sqrt{6^{31/16}} = 6^{31/32} \]

### Conclusion
Thus, the final value of the expression \( \sqrt{6\sqrt{6\sqrt{6\sqrt{6\sqrt{6}}}}} \) is:
\[ \boxed{6^{31/32}} \]