Solve each equation. See Example 7. x^2/3 = 2x^1/3

Verified step by step guidance

1. Start by rewriting the equation \(x^{2/3} = 2x^{1/3}\) to make it easier to work with. Notice that both sides involve powers of \(x\) with fractional exponents.
2. Introduce a substitution to simplify the equation. Let \(y = x^{1/3}\). Then, \(x^{2/3}\) can be written as \((x^{1/3})^2 = y^2\). Substitute these into the equation to get \(y^2 = 2y\).
3. Rewrite the equation \(y^2 = 2y\) in standard quadratic form by moving all terms to one side: \(y^2 - 2y = 0\).
4. Factor the quadratic equation: \(y(y - 2) = 0\). This gives two possible solutions for \(y\): \(y = 0\) or \(y = 2\).
5. Recall the substitution \(y = x^{1/3}\). Solve for \(x\) by cubing both sides of each solution: \(x = y^3\). So, \(x = 0^3\) and \(x = 2^3\). These give the solutions for \(x\).

Key Concepts

Exponents and Rational Exponents

Rational exponents represent roots and powers simultaneously, where a fractional exponent like x^(m/n) means the nth root of x raised to the mth power. Understanding how to manipulate and simplify expressions with rational exponents is essential for solving equations involving them.

Solving Equations by Substitution

When equations involve expressions like x^(1/3), substitution can simplify the problem. By letting a new variable equal x^(1/3), the equation becomes a polynomial in terms of the new variable, making it easier to solve before back-substituting to find x.

Solving Polynomial Equations

After substitution, the equation often reduces to a polynomial form. Knowing how to solve polynomial equations, including factoring and using the zero-product property, is crucial to find all possible solutions to the original equation.