Find the value of `sqrt(7sqrt(7sqrt(7)))`.

Answer

To find the value of \( \sqrt{7\sqrt{7\sqrt{7}}} \), we can follow these steps:

### Step 1: Define the expression
Let \( x = \sqrt{7\sqrt{7\sqrt{7}}} \).

### Step 2: Rewrite the expression
We can rewrite the expression inside the square root:
\[ x = \sqrt{7 \cdot \sqrt{7 \cdot \sqrt{7}}} \]
This means we have \( \sqrt{7} \) nested inside another square root.

### Step 3: Simplify the nested square root
Let’s express the inner square root:
\[ \sqrt{7\sqrt{7}} = \sqrt{7^{3/2}} = 7^{3/4} \]
Thus, we can rewrite \( x \) as:
\[ x = \sqrt{7 \cdot 7^{3/4}} = \sqrt{7^{1 + 3/4}} = \sqrt{7^{7/4}} \]

### Step 4: Simplify the square root
Now we can simplify:
\[ x = 7^{(7/4) \cdot (1/2)} = 7^{7/8} \]

### Step 5: Final answer
Thus, the value of \( \sqrt{7\sqrt{7\sqrt{7}}} \) is:
\[ \boxed{7^{7/8}} \]