The value of `sqrt(2 sqrt(2 sqrt(2...infty)))` is

Options

A. 0
B. 1
C. `2sqrt(2)`
D. 2

Answer

To solve the problem of finding the value of \( \sqrt{2 \sqrt{2 \sqrt{2 \ldots}}} \), we can follow these steps:

### Step 1: Define the Expression
Let \( x = \sqrt{2 \sqrt{2 \sqrt{2 \ldots}}} \).

### Step 2: Square Both Sides
To eliminate the square root, we square both sides:
\[ x^2 = 2 \sqrt{2 \sqrt{2 \ldots}} \]
Notice that the right side is still the same expression as \( x \):
\[ x^2 = 2x \]

### Step 3: Rearrange the Equation
Rearranging the equation gives us:
\[ x^2 - 2x = 0 \]

### Step 4: Factor the Equation
We can factor this equation:
\[ x(x - 2) = 0 \]

### Step 5: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x = 0 \)
2. \( x - 2 = 0 \) which leads to \( x = 2 \)

### Step 6: Determine the Valid Solution
Since \( x \) represents a square root and must be non-negative, we discard \( x = 0 \). Thus, the valid solution is:
\[ x = 2 \]

### Final Answer
The value of \( \sqrt{2 \sqrt{2 \sqrt{2 \ldots}}} \) is \( 2 \).

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