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

Verified step by step guidance

1. Start by rewriting the equation to clearly see the terms: \(\frac{(2x - 1)^2}{3} = x^{\frac{1}{3}}\). Note that the left side is the quantity \((2x - 1)\) squared, then divided by 3, and the right side is \(x\) raised to the one-third power.
2. To eliminate the fraction on the left side, multiply both sides of the equation by 3 to get rid of the denominator: \((2x - 1)^2 = 3x^{\frac{1}{3}}\).
3. Recognize that \(x^{\frac{1}{3}}\) is the cube root of \(x\). To simplify the equation, introduce a substitution: let \(y = x^{\frac{1}{3}}\), so that \(x = y^3\). Rewrite the equation in terms of \(y\).
4. Substitute \(x = y^3\) into the equation: \((2y^3 - 1)^2 = 3y\). This transforms the original equation into a polynomial equation in terms of \(y\).
5. Expand the left side: \((2y^3 - 1)^2 = (2y^3)^2 - 2 \times 2y^3 \times 1 + 1^2 = 4y^6 - 4y^3 + 1\). Set this equal to \$3y\( and write the equation as \)4y^6 - 4y^3 + 1 = 3y\(. Then, bring all terms to one side to form a polynomial equation: \)4y^6 - 4y^3 - 3y + 1 = 0\(. From here, solve for \)y\( using appropriate algebraic methods, then back-substitute to find \)x$.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers simultaneously. For example, x^(1/3) means the cube root of x, and x^(2/3) means the square of the cube root of x. Understanding how to manipulate and simplify expressions with rational exponents is essential for solving equations like the given one.

Equation Solving with Exponents

Solving equations involving exponents often requires isolating the variable term and applying inverse operations such as taking roots or raising both sides to a power. Recognizing when to raise both sides to a power to eliminate fractional exponents helps simplify the equation to a solvable form.

Checking for Extraneous Solutions

When solving equations with rational exponents, especially those involving even roots or fractional powers, extraneous solutions can arise. These are solutions that satisfy the transformed equation but not the original. Always substitute solutions back into the original equation to verify their validity.