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

Verified step by step guidance

1. Rewrite the equation \((x-2)^{2/3} = x^{1/3}\) to make it easier to work with. Notice that both sides have fractional exponents with denominator 3, so consider raising both sides to the power of 3 to eliminate the cube roots.
2. Raise both sides of the equation to the power of 3 to clear the fractional exponents: \(\left((x-2)^{2/3}\right)^3 = \left(x^{1/3}\right)^3\). Simplify the exponents by multiplying: \((x-2)^2 = x\).
3. Expand the left side of the equation: \((x-2)^2 = x^2 - 4x + 4\). So the equation becomes \(x^2 - 4x + 4 = x\).
4. Bring all terms to one side to set the equation equal to zero: \(x^2 - 4x + 4 - x = 0\), which simplifies to \(x^2 - 5x + 4 = 0\).
5. Solve the quadratic equation \(x^2 - 5x + 4 = 0\) by factoring, completing the square, or using the quadratic formula to find the possible values of \(x\). Remember to check each solution in the original equation because of the fractional exponents.

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 and Equating Expressions

To solve equations involving exponents, it is often necessary to isolate terms and rewrite the equation so that bases or exponents can be compared or equated. This may involve raising both sides to a power to eliminate fractional exponents or rewriting expressions with a common base.

Checking for Extraneous Solutions

When solving equations with rational exponents, especially involving even roots, some solutions may not satisfy the original equation. It is important to substitute solutions back into the original equation to verify their validity and exclude any extraneous solutions.