Find the value of `sqrt(2root(3)(4sqrt(2root(3)(4sqrt(2root(3)(4....))))))`.

Options

A. 2
B. `2^(2)`
C. `2^(3)`
D. `2^(5)`

Answer

To find the value of the expression \( \sqrt{2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \ldots}}}}}} \), we can follow these steps:

### Step 1: Define the Expression
Let \( x \) be the value of the entire expression:
\[ x = \sqrt{2 \sqrt[3]{4 x}} \]

### Step 2: Square Both Sides
To eliminate the square root, we square both sides:
\[ x^2 = 2 \sqrt[3]{4 x} \]

### Step 3: Isolate the Cube Root
Next, we can isolate the cube root term:
\[ \sqrt[3]{4 x} = \frac{x^2}{2} \]

### Step 4: Cube Both Sides
Now, we cube both sides to eliminate the cube root:
\[ 4 x = \left(\frac{x^2}{2}\right)^3 \]

### Step 5: Simplify the Right Side
Calculating the right side:
\[ \left(\frac{x^2}{2}\right)^3 = \frac{x^6}{8} \]
So, we have:
\[ 4 x = \frac{x^6}{8} \]

### Step 6: Multiply Both Sides by 8
To eliminate the fraction, we multiply both sides by 8:
\[ 32 x = x^6 \]

### Step 7: Rearrange the Equation
Rearranging gives us:
\[ x^6 - 32 x = 0 \]

### Step 8: Factor the Equation
Factoring out \( x \):
\[ x(x^5 - 32) = 0 \]

### Step 9: Solve for x
This gives us two solutions:
1. \( x = 0 \)
2. \( x^5 = 32 \)

For the second equation, taking the fifth root:
\[ x = 2 \]

### Conclusion
Thus, the value of the expression is:
\[ \boxed{2} \]