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

Verified step by step guidance

1. Start by making a substitution to simplify the equation. Let \(y = (x-1)^{\frac{1}{3}}\). This means \(y^2 = \left((x-1)^{\frac{1}{3}}\right)^2 = (x-1)^{\frac{2}{3}}\).
2. Rewrite the original equation in terms of \(y\): \(y^2 + y - 12 = 0\).
3. Recognize that this is a quadratic equation in \(y\). Use the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a=1\), \(b=1\), and \(c=-12\) to find the values of \(y\).
4. After finding the values of \(y\), substitute back to solve for \(x\) by setting \(y = (x-1)^{\frac{1}{3}}\). This means solving \(y^3 = x - 1\) for each value of \(y\).
5. Finally, solve for \(x\) by adding 1 to both sides: \(x = y^3 + 1\). This will give the solutions to the original equation.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers simultaneously, where an exponent like 2/3 means squaring and then taking the cube root. Understanding how to manipulate expressions with rational exponents is essential for simplifying and solving equations involving fractional powers.

Substitution Method

The substitution method involves replacing a complex expression with a single variable to simplify the equation. For example, letting y = (x-1)^(1/3) transforms the equation into a quadratic form, making it easier to solve.

Solving Quadratic Equations

Once the equation is rewritten in quadratic form, techniques like factoring, completing the square, or using the quadratic formula are used to find the variable's values. These solutions then need to be back-substituted to find the original variable x.