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

Options

A. `7^((31)/(64))`
B. `7^(1/(64))`
C. `7^((63)/(64))`
D. `7^((7)/(64))`

Answer

To solve the expression \( \sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7}}}}}} \), we can denote the entire expression as \( x \). Thus, we have:

\[ x = \sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7}}}}}} \]

### Step 1: Rewrite the expression
We can rewrite the expression in a more manageable form. Notice that the inner square root can also be represented as \( x \):

\[ x = \sqrt{7x} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:

\[ x^2 = 7x \]

### Step 3: Rearrange the equation
Rearranging this equation gives us:

\[ x^2 - 7x = 0 \]

### Step 4: Factor the equation
We can factor out \( x \):

\[ x(x - 7) = 0 \]

### Step 5: Solve for \( x \)
Setting each factor equal to zero gives us:

\[ x = 0 \quad \text{or} \quad x = 7 \]

Since \( x \) represents a square root and must be non-negative, we discard \( x = 0 \). Therefore, we have:

\[ x = 7 \]

### Final Answer
Thus, the value of \( \sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7}}}}}} \) is:

\[ \boxed{7} \]