What is the value of
`sqrt(5sqrt(5sqrt(5sqrt(....oo))))`

Options

A. 5
B. `sqrt(5)`
C. 1
D. `(5)^(1//4)`

Explanation

To find the value of the expression \( \sqrt{5\sqrt{5\sqrt{5\sqrt{\ldots}}}} \), we can follow these steps:

### Step 1: Define the expression
Let \( x = \sqrt{5\sqrt{5\sqrt{5\sqrt{\ldots}}}} \). This means that the entire expression can be represented as \( x \).

### Step 2: Square both sides
To eliminate the square root, we square both sides:
\[ x^2 = 5\sqrt{5\sqrt{5\sqrt{\ldots}}} \]
Since the right side is the same as our original expression, we can rewrite it as:
\[ x^2 = 5x \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 - 5x = 0 \]

### Step 4: Factor the equation
Factoring out \( x \) from the left side:
\[ x(x - 5) = 0 \]

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

### Step 6: Determine the valid solution
Since \( x \) represents a square root, it must be non-negative. Therefore, we discard \( x = 0 \) as a valid solution. The only valid solution is:
\[ x = 5 \]

### Final Answer
The value of \( \sqrt{5\sqrt{5\sqrt{5\sqrt{\ldots}}}} \) is \( 5 \).

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