Solve each equation. x^-2/3+x^-1/3-6=0

Verified step by step guidance

1. Recognize that the equation involves negative fractional exponents: \(x^{-\frac{2}{3}} + x^{-\frac{1}{3}} - 6 = 0\). To simplify, introduce a substitution to rewrite the equation in terms of a single variable. Let \(y = x^{-\frac{1}{3}}\).
2. Rewrite each term in the equation using the substitution \(y = x^{-\frac{1}{3}}\). Note that \(x^{-\frac{2}{3}} = (x^{-\frac{1}{3}})^2 = y^2\). So the equation becomes \(y^2 + y - 6 = 0\).
3. Solve the quadratic equation \(y^2 + y - 6 = 0\) by factoring, completing the square, or using the quadratic formula. This will give you the possible values for \(y\).
4. After finding the values of \(y\), recall that \(y = x^{-\frac{1}{3}} = \frac{1}{x^{\frac{1}{3}}}\). Rewrite this as \(x^{\frac{1}{3}} = \frac{1}{y}\) to solve for \(x\).
5. Finally, solve for \(x\) by cubing both sides: \(x = \left( \frac{1}{y} \right)^3\). Calculate this for each value of \(y\) found in step 3 to get the solutions for \(x\).

Key Concepts

Rational Exponents

Rational exponents represent roots and powers combined, where an exponent like x^(m/n) means the nth root of x raised to the mth power. Understanding how to manipulate expressions with rational exponents is essential for rewriting and simplifying the given equation.

Substitution Method

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

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically solved by factoring, completing the square, or using the quadratic formula. After substitution, the equation becomes quadratic, so knowing how to solve it is crucial.