(x-3)^2/5=(4x)^1/5

3 min read Jun 17, 2024

Solving the Equation: (x-3)^2/5 = (4x)^1/5

This article will guide you through the process of solving the equation (x-3)^2/5 = (4x)^1/5. We'll utilize algebraic manipulation and techniques to arrive at the solution.

Step 1: Isolate the Exponents

To simplify the equation, let's get rid of the fractional exponents. We can achieve this by raising both sides of the equation to the power of 5:

[(x-3)^2/5]^5 = [(4x)^1/5]^5

This simplifies to:

(x-3)^2 = 4x

Step 2: Expand and Rearrange

Now, expand the left side of the equation and move all terms to one side:

x^2 - 6x + 9 = 4x

x^2 - 10x + 9 = 0

Step 3: Factor the Quadratic Equation

The equation is now a quadratic equation. We can factor it to find its roots:

(x-9)(x-1) = 0

Therefore, the solutions are:

x = 9 and x = 1

Step 4: Verify the Solutions

It's crucial to check if the solutions we obtained are valid. We can do this by plugging each solution back into the original equation.

For x = 9:

[(9-3)^2/5] = [(4*9)^1/5] [6^2/5] = [36^1/5] 36^1/5 = 36^1/5 (This is true)

For x = 1:

[(1-3)^2/5] = [(4*1)^1/5] [(-2)^2/5] = [4^1/5] 4^1/5 = 4^1/5 (This is true)

Both solutions satisfy the original equation.

Conclusion

The equation (x-3)^2/5 = (4x)^1/5 has two solutions: x = 9 and x = 1. We arrived at these solutions by isolating the exponents, expanding the equation, factoring the quadratic, and finally verifying the solutions.