To solve the equation \((2x - 1)^7 = 3^{14}\), we need to isolate \( x \). Here's the step-by-step solution:

### Step 1: Take the seventh root of both sides

To remove the exponent on the left side, take the seventh root of both sides of the equation:

\[
(2x - 1) = (3^{14})^{\frac{1}{7}}
\]

Simplify the right side:

\[
(2x - 1) = 3^{\frac{14}{7}}
\]

\[
(2x - 1) = 3^2
\]

\[
2x - 1 = 9
\]

### Step 2: Solve for \( x \)

Add 1 to both sides:

\[
2x = 9 + 1
\]

\[
2x = 10
\]

Divide by 2:

\[
x = 5
\]

### Final Answer:

\[
\boxed{x = 5}
\]

Would you like any additional details or have any questions?

### Related Questions:

1. What is the general method for solving exponential equations like this one?
2. How can you solve equations involving fractional exponents?
3. What happens if the bases on both sides of an equation are not equal?
4. How would you approach this problem if the equation were \((2x - 1)^7 = 3^{13}\)?
5. How can logarithms be used to solve exponential equations?

**Tip:** When solving equations with exponents, isolating the exponential expression and using roots or logarithms are key strategies.

Learn how to solve the exponential equation (2x-1)^7 = 3^14 step-by-step. This solution involves exponentiation, taking roots, and isolating the variable. Perfect for students in Grades 9-11 studying advanced algebra.

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