Solve each equation. See Examples 8 and 9. (2x-1)^2/3+2(2x-1)^1/3-3=0

Verified step by step guidance

1. Start by making a substitution to simplify the equation. Let \(y = (2x - 1)^{\frac{1}{3}}\). This means \(y^2 = (2x - 1)^{\frac{2}{3}}\).
2. Rewrite the original equation in terms of \(y\): \(y^2 + 2y - 3 = 0\).
3. Recognize that this is a quadratic equation in \(y\). Use the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a=1\), \(b=2\), and \(c=-3\) to find the values of \(y\).
4. After finding the values of \(y\), substitute back \(y = (2x - 1)^{\frac{1}{3}}\) to get equations of the form \((2x - 1)^{\frac{1}{3}} = y\).
5. Solve each resulting equation by cubing both sides to eliminate the cube root, then solve the resulting linear equation for \(x\).

Key Concepts

Radical Exponents and Fractional Powers

Radical exponents represent roots, where a fractional exponent like 1/3 denotes a cube root. Understanding how to rewrite and manipulate expressions with fractional exponents is essential for simplifying and solving equations involving roots.

Substitution Method for Solving Equations

Substitution involves replacing a complex expression with a single variable to simplify the equation. This technique is useful when dealing with powers or roots, allowing the equation to be transformed into a more familiar quadratic or polynomial form.

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two and can be solved by factoring, completing the square, or using the quadratic formula. Recognizing a quadratic form after substitution helps in finding the roots of the original equation.