The value of `sqrt(2^3sqrt(4sqrt(2^3sqrt(4.........))))` is-

Options

A. 2
B. `2^2`
C. `2^4`
D. `2^5`

Answer

To solve the expression \( \sqrt{2^3 \sqrt{4 \sqrt{2^3 \sqrt{4 \ldots}}}} \), let's denote the entire expression as \( x \).

1. **Set up the equation**:
\[ x = \sqrt{2^3 \sqrt{4 \sqrt{2^3 \sqrt{4 \ldots}}}} \]
Notice that the inner expression is the same as \( x \). Therefore, we can rewrite the equation as:
\[ x = \sqrt{2^3 \sqrt{4x}} \]

2. **Simplify the equation**:
We can simplify \( \sqrt{4x} \) as \( 2\sqrt{x} \). Thus, we have:
\[ x = \sqrt{2^3 \cdot 2 \sqrt{x}} = \sqrt{2^4 \sqrt{x}} = \sqrt{16 \sqrt{x}} \]

3. **Square both sides**:
To eliminate the square root, we square both sides:
\[ x^2 = 16 \sqrt{x} \]

4. **Rearrange the equation**:
Rearranging gives us:
\[ x^2 - 16\sqrt{x} = 0 \]

5. **Substitute \( y = \sqrt{x} \)**:
Let \( y = \sqrt{x} \). Then \( x = y^2 \), and substituting gives:
\[ y^4 - 16y = 0 \]

6. **Factor the equation**:
Factoring out \( y \) gives:
\[ y(y^3 - 16) = 0 \]
This gives us two possible solutions: \( y = 0 \) or \( y^3 = 16 \).

7. **Solve for \( y \)**:
From \( y^3 = 16 \), we find:
\[ y = \sqrt[3]{16} = 2^{4/3} \]

8. **Find \( x \)**:
Since \( x = y^2 \), we have:
\[ x = (2^{4/3})^2 = 2^{8/3} \]

9. **Final answer**:
The value of \( x \) is \( 2^{8/3} \).

Now, to express \( 2^{8/3} \) in a more recognizable form, we can rewrite it as:
\[ 2^{8/3} = \sqrt[3]{2^8} = \sqrt[3]{256} \]

Thus, the final answer is:
\[ \sqrt[3]{256} \]