## Solving the Equation (x^2-8x+15)^3x-1=(x^2-8x+15)^x+3

This equation involves exponents with expressions as bases. To solve it, we'll utilize the following key principles:

**1. Equating Exponents:** If we have two powers with the same base and they are equal, their exponents must also be equal.

**2. Factoring and Solving Equations:** We can often simplify equations by factoring or rearranging terms.

Let's break down the solution step-by-step:

**Step 1: Simplify the equation**

Notice that both sides of the equation have the same base: (x^2 - 8x + 15). We can rewrite the equation to make it easier to work with:

(x^2 - 8x + 15)^(3x-1) = (x^2 - 8x + 15)^(x+3)

**Step 2: Apply the principle of equating exponents**

Since the bases are the same, we can equate the exponents:

3x - 1 = x + 3

**Step 3: Solve for x**

Now we have a simple linear equation. Let's solve for x:

- 3x - x = 3 + 1
- 2x = 4
- x = 2

**Step 4: Verify the Solution**

It's always a good idea to check our solution by plugging it back into the original equation:

- (2^2 - 8 * 2 + 15)^(3 * 2 - 1) = (2^2 - 8 * 2 + 15)^(2 + 3)
- (4 - 16 + 15)^(5) = (4 - 16 + 15)^(5)
- 3^5 = 3^5

The equation holds true, so our solution **x = 2** is correct.

**Therefore, the solution to the equation (x^2-8x+15)^3x-1=(x^2-8x+15)^x+3 is x = 2.**