Value of `sqrt(7sqrt(7sqrt(7sqrt(7))))`......

Options

A. 7
B. 3
C. 4
D. 2

Answer

To solve the expression \( \sqrt{7\sqrt{7\sqrt{7\sqrt{7}}}} \), we can use the concept of infinite nesting of square roots. Let's denote the entire expression as \( x \).

### Step-by-step Solution:

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

2. **Recognize the pattern**:
Notice that the expression inside the square root is the same as \( x \). Thus, we can rewrite the equation as:
\[ x = \sqrt{7x} \]

3. **Square both sides**:
To eliminate the square root, we square both sides of the equation:
\[ x^2 = 7x \]

4. **Rearrange the equation**:
Move all terms to one side to form a standard quadratic equation:
\[ x^2 - 7x = 0 \]

5. **Factor the equation**:
Factor out \( x \):
\[ x(x - 7) = 0 \]

6. **Solve for \( x \)**:
Set each factor equal to zero:
\[ x = 0 \quad \text{or} \quad x = 7 \]

7. **Determine the valid solution**:
Since \( x \) represents a square root and must be non-negative, we discard \( x = 0 \). Thus, we have:
\[ x = 7 \]

### Final Answer:
The value of \( \sqrt{7\sqrt{7\sqrt{7\sqrt{7}}}} \) is \( 7 \).