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

Verified step by step guidance

1. Start by letting $y=x^{1/3}$. This substitution simplifies the equation to $(y^2-2)=y$.
2. Rearrange the equation to form a quadratic equation: $y^2-y-2=0$.
3. Factor the quadratic equation: $(y-2)(y+1)=0$.
4. Set each factor equal to zero and solve for $y$: $y-2=0$ or $y+1=0$.
5. Substitute back $y=x^{1/3}$ to find $x$ for each solution of $y$.

Key Concepts

Exponents and Radicals

Exponents represent repeated multiplication of a number, while radicals are the inverse operation, indicating the root of a number. In the given equation, both sides involve fractional exponents, which can be rewritten in radical form. Understanding how to manipulate these forms is crucial for solving equations involving powers and roots.

Equations and Solutions

An equation is a mathematical statement asserting the equality of two expressions. To solve an equation, one must isolate the variable, often by performing inverse operations. In this case, manipulating the equation to find the values of x that satisfy the equality is essential for finding the solution.

Squaring and Square Roots

Squaring a number means multiplying it by itself, while taking the square root is finding a number that, when squared, gives the original number. In the context of the equation, understanding how to apply these operations correctly is vital, especially since squaring can introduce extraneous solutions that must be checked against the original equation.