Find the value of `sqrt(9sqrt(9sqrt(9.......oo)))`.

Options

A. 81
B. 3
C. 9
D. `oo`

Answer

To solve the expression \( \sqrt{9\sqrt{9\sqrt{9\ldots}}} \), we can follow these steps:

### Step 1: Define the Expression
Let \( x = \sqrt{9\sqrt{9\sqrt{9\ldots}}} \). This means that the entire expression can be represented as \( x \).

### Step 2: Rewrite the Expression
Since the expression inside the square root is the same as \( x \), we can rewrite it as:
\[ x = \sqrt{9x} \]

### Step 3: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ x^2 = 9x \]

### Step 4: Rearrange the Equation
Rearranging the equation gives us:
\[ x^2 - 9x = 0 \]

### Step 5: Factor the Equation
We can factor out \( x \):
\[ x(x - 9) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us two possible solutions:
1. \( x = 0 \)
2. \( x - 9 = 0 \) which leads to \( x = 9 \)

### Step 7: Determine the Valid Solution
Since we are dealing with a square root and the expression represents a length (or a value), we discard \( x = 0 \) because it does not make sense in this context. Therefore, we have:
\[ x = 9 \]

### Conclusion
The value of \( \sqrt{9\sqrt{9\sqrt{9\ldots}}} \) is \( 9 \).