Let's solve the equation:

\[
(1.06)^2 = \left( \frac{1 + x}{4} \right)^4
\]

### Step 1: Simplify the left-hand side
First, calculate \( (1.06)^2 \):

\[
1.06^2 = 1.1236
\]

Thus, the equation becomes:

\[
1.1236 = \left( \frac{1 + x}{4} \right)^4
\]

### Step 2: Take the fourth root of both sides
To isolate \( \frac{1 + x}{4} \), take the fourth root of both sides:

\[
\sqrt[4]{1.1236} = \frac{1 + x}{4}
\]

Calculating \( \sqrt[4]{1.1236} \):

\[
\sqrt[4]{1.1236} \approx 1.02961
\]

Now, the equation becomes:

\[
1.02961 = \frac{1 + x}{4}
\]

### Step 3: Solve for \( x \)
Multiply both sides by 4:

\[
1 + x = 4 \times 1.02961
\]
\[
1 + x = 4.11844
\]

Now subtract 1 from both sides:

\[
x = 4.11844 - 1
\]
\[
x \approx 3.11844
\]

Thus, the solution is:

\[
x \approx 3.1184
\]

Would you like any additional details or have further questions?

### Related questions:
1. How do you solve other types of exponential equations?
2. Can this method be applied to equations involving square roots?
3. What if the exponent on the right side were different, like 3 or 5?
4. How do you solve exponential equations where both sides have different bases?
5. How does this method apply to logarithmic equations?

**Tip:** When dealing with exponents, it's often helpful to take the logarithm if you're stuck with different bases.

Learn how to solve the exponential equation (1.06)^2 = ((1+x)/4)^4 with a step-by-step solution. This problem covers key algebraic concepts, including solving for variables inside exponents and roots. Suitable for high school students in Grades 9-11.

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