( sqrt( sqrt( sqrt( sqrt(5+2^(x))))))/(2^(x)-1) = 1/3^( 7/8 )

We are tasked with solving the equation:

\[
\frac{\sqrt{\sqrt{\sqrt{\sqrt{5 + 2^x}}}}}{2^x - 1} = \frac{1}{3^{7/8}}
\]

### Step-by-step solution:

1. **Simplify both sides:**
   Start by rewriting the equation:

   \[
   \frac{\left(5 + 2^x\right)^{1/16}}{2^x - 1} = \frac{1}{3^{7/8}}
   \]

   The left-hand side (LHS) is \( (5 + 2^x)^{1/16} \) because the square root is applied four times (which is equivalent to taking the 16th root of the expression).

2. **Raise both sides to the 16th power:**
   To eliminate the fractional exponents on the left-hand side, raise both sides of the equation to the 16th power:

   \[
   \left( \frac{(5 + 2^x)^{1/16}}{2^x - 1} \right)^{16} = \left( \frac{1}{3^{7/8}} \right)^{16}
   \]

   This simplifies to:

   \[
   \frac{5 + 2^x}{(2^x - 1)^{16}} = \frac{1}{3^{14}}
   \]

3. **Cross-multiply:**

   \[
   5 + 2^x = \frac{(2^x - 1)^{16}}{3^{14}}
   \]

   Now we have a more complex equation. Solving this exactly will involve checking for values of \( x \) that make both sides equal.

### Trial and Error for \( x \):

Let's try small values for \( x \):

- **For \( x = 1 \):**

   \[
   5 + 2^1 = 5 + 2 = 7
   \]
   The right-hand side becomes:
   \[
   \frac{(2^1 - 1)^{16}}{3^{14}} = \frac{(2 - 1)^{16}}{3^{14}} = \frac{1^{16}}{3^{14}} = \frac{1}{3^{14}}
   \]
   Clearly, \( 7 \neq \frac{1}{3^{14}} \), so \( x = 1 \) is not a solution.

- **For \( x = 2 \):**

   \[
   5 + 2^2 = 5 + 4 = 9
   \]
   The right-hand side becomes:
   \[
   \frac{(2^2 - 1)^{16}}{3^{14}} = \frac{(4 - 1)^{16}}{3^{14}} = \frac{3^{16}}{3^{14}} = 3^2 = 9
   \]
   This is a match.

Thus, \( x = 2 \) is the solution.

### Final answer:
\[
x = 2
\]

Would you like further details or have any questions?

Here are five related questions for further exploration:

1. How would the solution change if the exponent on the denominator was different?
2. What happens if we replace \( 5 + 2^x \) with \( 3 + 2^x \)?
3. Can you generalize the method used to solve similar nested radical equations?
4. What would happen if the right-hand side was \( \frac{1}{4^{7/8}} \)?
5. How would you graph the function \( y = \frac{\sqrt{\sqrt{\sqrt{\sqrt{5 + 2^x}}}}}{2^x - 1} \)?

**Tip:** Always try small integer values for \( x \) when solving such exponential equations—it can lead to the solution quickly!

sqrt( sqrt( sqrt( sqrt(5+2^(x))))))/(2^(x)-1) = 1/3^( 7/8 )

Explore how to solve a nested radical equation with exponential expressions: sqrt(sqrt(5+2^x))/(2^x - 1) = 1/3^(7/8). This problem involves simplifying radicals, using properties of exponents, and trial and error methods for solving complex algebraic equations. Suitable for students in Grades 10-12.

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