The value of `sqrt(5 sqrt(5 sqrt(5)...))` is

Options

A. 1
B. 2.5
C. 5
D. 25

Answer

To solve the problem of finding the value of \( \sqrt{5 \sqrt{5 \sqrt{5 \ldots}}} \), we can follow these steps:

### Step 1: Define the expression
Let \( x = \sqrt{5 \sqrt{5 \sqrt{5 \ldots}}} \). This means that \( x \) is equal to the entire expression.

### Step 2: Square both sides
To eliminate the square root, we square both sides of the equation:
\[ x^2 = 5 \sqrt{5 \sqrt{5 \ldots}} \]
Notice that the right side is the same as \( x \):
\[ x^2 = 5x \]

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

### Step 4: Factor the equation
We can factor the left side:
\[ x(x - 5) = 0 \]

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

### Step 6: Determine the valid solution
Since \( x \) represents a length (as it is derived from a square root), we discard the negative solution \( x = 0 \). Thus, the valid solution is:
\[ x = 5 \]

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

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