The value of `sqrt(2root3(4sqrt(2root3(4sqrt(2root3(4............))))))` is

Options

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

Answer

To solve 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:
\[ x = \sqrt{2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \sqrt{2 \sqrt[3]{4 \ldots}}}}}} \]

### Step 2: Rewrite the Expression
Notice that the expression inside the square root repeats itself. Therefore, we can rewrite it as:
\[ x = \sqrt{2 \sqrt[3]{4x}} \]

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

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

### Step 5: Cube Both Sides
Now, cube both sides to eliminate the cube root:
\[ 4x = \left(\frac{x^2}{2}\right)^3 \]
Calculating the right side:
\[ 4x = \frac{x^6}{8} \]

### Step 6: Simplify the Equation
Multiply both sides by 8 to eliminate the fraction:
\[ 32x = x^6 \]

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

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

### Step 9: Solve for \( x \)
This gives us two solutions:
1. \( x = 0 \) (not valid in this context)
2. \( x^5 = 32 \)

Taking the fifth root of both sides:
\[ x = 2 \]

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

---