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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.