Solve each equation. See Example 7. (2x-1)^2/3 = x^1/3

Verified step by step guidance

1. Start by letting \( y = x^{1/3} \). This substitution simplifies the equation to \((2y^3 - 1)^{2/3} = y\).
2. Raise both sides of the equation to the power of 3 to eliminate the fractional exponent: \((2y^3 - 1)^2 = y^3\).
3. Expand the left side of the equation: \(4y^6 - 4y^3 + 1 = y^3\).
4. Rearrange the equation to set it to zero: \(4y^6 - 5y^3 + 1 = 0\).
5. Let \( z = y^3 \), transforming the equation into a quadratic form: \(4z^2 - 5z + 1 = 0\). Solve this quadratic equation for \( z \), and then back-substitute to find \( y \) and ultimately \( x \).

Key Concepts

Exponents and Radicals

Understanding exponents and radicals is crucial for solving equations involving powers and roots. In this equation, the term (2x-1) is raised to the power of 2/3, which indicates both squaring the expression and taking the cube root. Similarly, x^1/3 represents the cube root of x. Mastery of these concepts allows for proper manipulation and simplification of the equation.

Isolating Variables

Isolating variables is a fundamental technique in algebra that involves rearranging an equation to solve for a specific variable. In this case, you will need to manipulate the equation to isolate x. This often includes applying inverse operations, such as raising both sides to a power or moving terms across the equation, which is essential for finding the solution.

Checking Solutions

After solving an equation, it is important to check the solutions to ensure they satisfy the original equation. This step helps identify any extraneous solutions that may arise from the manipulation of the equation, especially when dealing with roots and powers. Verifying solutions ensures accuracy and confirms that the derived values are valid within the context of the problem.