Find the value of `sqrt(5sqrt(5sqrt(5sqrt(5))))`.

Options

A. `5^((1)/(16))`
B. `5^((15)/(32))`
C. `5^((15)/(16))`
D. `5^((1)/(32))`

Answer

To solve the expression \( \sqrt{5\sqrt{5\sqrt{5\sqrt{5}}}} \), we can simplify it step by step.

### Step 1: Rewrite the expression
Let's denote the expression as \( x \):
\[ x = \sqrt{5\sqrt{5\sqrt{5\sqrt{5}}}} \]

### Step 2: Simplify the innermost square root
We can start simplifying from the innermost square root:
\[ x = \sqrt{5\sqrt{5\sqrt{5\sqrt{5}}}} = \sqrt{5\sqrt{5\sqrt{5x}}} \]

### Step 3: Substitute and simplify further
Now, we can replace \( \sqrt{5\sqrt{5\sqrt{5}}} \) with \( \sqrt{5x} \):
\[ x = \sqrt{5\sqrt{5x}} \]

### Step 4: Simplify the next square root
Next, we simplify \( \sqrt{5x} \):
\[ x = \sqrt{5\sqrt{5x}} = \sqrt{5 \cdot 5^{1/2} \cdot x^{1/2}} = \sqrt{5^{3/2} \cdot x^{1/2}} \]

### Step 5: Rewrite the expression in terms of powers
This can be rewritten as:
\[ x = (5^{3/2} \cdot x^{1/2})^{1/2} = 5^{3/4} \cdot x^{1/4} \]

### Step 6: Isolate \( x \)
Now, we can isolate \( x \):
\[ x^{3/4} = 5^{3/4} \]

### Step 7: Raise both sides to the power of \( \frac{4}{3} \)
To solve for \( x \), we raise both sides to the power of \( \frac{4}{3} \):
\[ x = (5^{3/4})^{4/3} = 5^{(3/4) \cdot (4/3)} = 5^{1} = 5 \]

### Final Answer
Thus, the value of \( \sqrt{5\sqrt{5\sqrt{5\sqrt{5}}}} \) is:
\[ \boxed{5} \]