`sqrt(3sqrt(3sqrt(3......)))` is equal to

Options

A. `sqrt(3)`
B. `3`
C. `2sqrt(3)`
D. `3sqrt(3)`

Answer

To solve the expression \( \sqrt{3\sqrt{3\sqrt{3\ldots}}} \), we can denote it as \( x \). Thus, we can write:

\[ x = \sqrt{3\sqrt{3\sqrt{3\ldots}}} \]

Since the expression inside the square root is the same as \( x \), we can rewrite the equation as:

\[ x = \sqrt{3x} \]

Next, we will square both sides to eliminate the square root:

\[ x^2 = 3x \]

Now, we will rearrange this equation to bring all terms to one side:

\[ x^2 - 3x = 0 \]

Next, we can factor the left-hand side:

\[ x(x - 3) = 0 \]

This gives us two possible solutions:

1. \( x = 0 \)
2. \( x = 3 \)

Since we are dealing with square roots and the expression represents a positive quantity, we discard \( x = 0 \). Therefore, we have:

\[ x = 3 \]

Thus, the value of \( \sqrt{3\sqrt{3\sqrt{3\ldots}}} \) is equal to \( 3 \).