Solve each equation. (x-2)^2/3 = x^1/3

Verified step by step guidance

1. Start by rewriting the equation \(\frac{(x-2)^2}{3} = x^{\frac{1}{3}}\) carefully. Note that the left side means \(\left((x-2)^2\right)^{\frac{1}{3}}\), which is the cube root of \((x-2)^2\).
2. To eliminate the fractional exponents, raise both sides of the equation to the power of 3 to get rid of the cube roots. This gives: \(\left(\left((x-2)^2\right)^{\frac{1}{3}}\right)^3 = \left(x^{\frac{1}{3}}\right)^3\).
3. Simplify both sides using the property \((a^{m})^{n} = a^{mn}\). The left side becomes \((x-2)^2\) and the right side becomes \(x\).
4. Now you have the equation \((x-2)^2 = x\). Expand the left side using the formula \((a-b)^2 = a^2 - 2ab + b^2\) to get \(x^2 - 4x + 4 = x\).
5. Bring all terms to one side to form a quadratic equation: \(x^2 - 4x + 4 - x = 0\), which simplifies to \(x^2 - 5x + 4 = 0\). Then solve this quadratic equation using factoring, completing the square, or the quadratic formula.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers simultaneously. For example, x^(m/n) means the n-th root of x raised to the m-th power. Understanding how to manipulate and simplify expressions with rational exponents is essential for solving equations like (x-2)^(2/3) = x^(1/3).

Isolating Terms and Substitution

To solve equations involving rational exponents, it is often helpful to isolate terms or use substitution. For instance, letting y = x^(1/3) can simplify the equation into a polynomial form, making it easier to solve. This technique reduces complexity and helps find solutions systematically.

Checking for Extraneous Solutions

When solving equations with rational exponents, especially involving even roots, some solutions may not satisfy the original equation. These extraneous solutions arise from the process of raising both sides to powers. Always substitute solutions back into the original equation to verify their validity.