**Solving the Equation: 625^x = (1/25)^x + 2**

In this article, we will explore the solution to the equation 625^x = (1/25)^x + 2. This equation involves exponential functions and requires some algebraic manipulations to solve.

**Step 1: Simplify the equation**

Let's start by simplifying the right-hand side of the equation. We can rewrite (1/25)^x as:

(1/25)^x = (25^-1)^x = 25^(-x)

So, the equation becomes:

625^x = 25^(-x) + 2

**Step 2: Express both sides in terms of 5**

Notice that 625 can be expressed as 5^4, and 25 can be expressed as 5^2. Let's rewrite the equation in terms of 5:

(5^4)^x = 5^(2(-x)) + 2

Using the property of exponents, we can simplify the left-hand side:

5^(4x) = 5^(2(-x)) + 2

**Step 3: Equate the exponents**

Since the bases are the same (both are 5), we can equate the exponents:

4x = -2x

Simplifying the equation, we get:

6x = 0

**Step 4: Solve for x**

The solution to the equation is:

x = 0

Therefore, the value of x that satisfies the equation 625^x = (1/25)^x + 2 is x = 0.

**Conclusion**

In this article, we have solved the equation 625^x = (1/25)^x + 2 using algebraic manipulations and exponential functions. The solution to the equation is x = 0.