Solve each equation in Exercises 41–60 by making an appropriate substitution. $x^{\frac{2}{5}}+x^{\frac{1}{5}}-6=0$

Verified step by step guidance

1. Identify the substitution to simplify the equation. Notice that the exponents are fractional and related: \(x^{2/5}\) and \(x^{1/5}\). Let \(u = x^{1/5}\), so that \(u^2 = x^{2/5}\).
2. Rewrite the original equation in terms of \(u\): replace \(x^{2/5}\) with \(u^2\) and \(x^{1/5}\) with \(u\). The equation becomes \(u^2 + u - 6 = 0\).
3. Solve the quadratic equation \(u^2 + u - 6 = 0\) using factoring, completing the square, or the quadratic formula. This will give you the possible values of \(u\).
4. After finding the values of \(u\), recall that \(u = x^{1/5}\). To find \(x\), raise both sides of the equation \(u = x^{1/5}\) to the 5th power, resulting in \(x = u^5\).
5. Calculate \(x\) for each value of \(u\) found in step 3 by computing \(x = u^5\). These are the solutions to the original equation.

Key Concepts

Exponents and Rational Powers

Rational exponents represent roots and powers, such as x^(m/n) meaning the n-th root of x raised to the m-th power. Understanding how to manipulate and simplify expressions with fractional exponents is essential for rewriting and solving equations involving these terms.

Substitution Method

Substitution involves replacing a complex expression with a simpler variable to transform the equation into a more familiar form, often a polynomial. This technique simplifies solving equations that contain complicated terms, such as fractional powers, by reducing them to quadratic or linear equations.

Solving Quadratic Equations

Once substitution reduces the equation to a quadratic form, solving it involves finding values of the variable that satisfy the equation using factoring, completing the square, or the quadratic formula. Understanding how to solve quadratics is crucial to find the solutions to the original equation after back-substitution.