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. Let $u=(2x-1{)}^{1/3}$. This substitution simplifies the equation to $u^2+2u-3=0$.
2. Recognize that $u^2+2u-3=0$ is a quadratic equation. Factor it to find $(u+3)(u-1)=0$.
3. Set each factor equal to zero: $u+3=0$ and $u-1=0$. Solve for $u$ to get $u=-3$ and $u=1$.
4. Substitute back $u=(2x-1{)}^{1/3}$ into the solutions for $u$. This gives two equations: $(2x-1{)}^{1/3}=-3$ and $(2x-1{)}^{1/3}=1$.
5. Solve each equation for $x$: For $(2x-1{)}^{1/3}=-3$, cube both sides to get $2x-1=-27$. For $(2x-1{)}^{1/3}=1$, cube both sides to get $2x-1=1$. Then solve each resulting linear equation for $x$.

Key Concepts

Exponents and Radicals

Exponents represent repeated multiplication of a base number, while radicals involve roots of numbers. In the given equation, the term (2x-1) is raised to the power of 2/3, indicating both squaring and taking the cube root. Understanding how to manipulate these expressions is crucial for solving the equation.

Substitution Method

The substitution method involves replacing a complex expression with a simpler variable to make solving easier. In this case, letting y = (2x-1)^(1/3) simplifies the equation significantly, transforming it into a polynomial form that is easier to solve.

Polynomial Equations

Polynomial equations are mathematical expressions that involve variables raised to whole number powers. The equation derived from the substitution will be a polynomial, which can be solved using techniques such as factoring, the quadratic formula, or numerical methods, depending on its degree.