Solve each equation. See Examples 8 and 9. (x-1)^2/3+(x-1)^1/3 -12 = 0

Verified step by step guidance

1. Let us start by making a substitution to simplify the equation. Set \( y = (x-1)^{1/3} \). This means \( y^2 = (x-1)^{2/3} \).
2. Substitute \( y \) and \( y^2 \) into the equation: \( y^2 + y - 12 = 0 \).
3. Now, we have a quadratic equation in terms of \( y \). To solve it, we can use the quadratic formula: \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -12 \).
4. Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots.
5. Once you find the values of \( y \), substitute back \( y = (x-1)^{1/3} \) to solve for \( x \) by cubing both sides and solving for \( x \).

Key Concepts

Exponents and Radicals

Understanding exponents and radicals is crucial for solving equations involving powers and roots. In this equation, the terms (x-1) are raised to fractional exponents, which represent roots. For example, (x-1)^(2/3) indicates the square of the cube root of (x-1). Mastery of these concepts allows for proper manipulation and simplification of the equation.

Substitution Method

The substitution method is a powerful technique for simplifying complex equations. In this case, letting u = (x-1)^(1/3) transforms the original equation into a more manageable polynomial form. This approach can help reduce the complexity of the problem, making it easier to solve for x after finding u.

Polynomial Equations

Polynomial equations are expressions that equate a polynomial to zero. The equation derived from the substitution method will typically be a cubic polynomial, which can be solved using various methods such as factoring, synthetic division, or the Rational Root Theorem. Understanding how to work with polynomial equations is essential for finding the roots and solutions of the original equation.