Solve each equation. See Example 7. (3x+7)^1/3-(4x+2)^1/3=0

Verified step by step guidance

1. Start with the given equation: \(\left(3x+7\right)^{\frac{1}{3}} - \left(4x+2\right)^{\frac{1}{3}} = 0\).
2. Isolate one of the cube root expressions by adding \(\left(4x+2\right)^{\frac{1}{3}}\) to both sides: \(\left(3x+7\right)^{\frac{1}{3}} = \left(4x+2\right)^{\frac{1}{3}}\).
3. Since the cube root functions are equal, set their radicands equal to each other: \(3x + 7 = 4x + 2\).
4. Solve the resulting linear equation for \(x\) by subtracting \$3x$ from both sides and subtracting 2 from both sides: \(7 - 2 = 4x - 3x\).
5. Simplify and solve for \(x\): \(5 = x\), so \(x = 5\). Remember to check this solution by substituting it back into the original equation to verify it does not produce any extraneous roots.

Key Concepts

Radical Equations

Radical equations involve variables within roots, such as cube roots or square roots. Solving them often requires isolating the radical expression and then eliminating the root by raising both sides to the appropriate power. Care must be taken to check for extraneous solutions introduced during this process.

Cube Roots and Their Properties

A cube root of a number x is a value that, when cubed, gives x. Unlike square roots, cube roots are defined for all real numbers, including negatives. Understanding how to manipulate and simplify expressions with cube roots is essential for solving equations involving fractional exponents like 1/3.

Equation Solving by Substitution

Substitution involves replacing complex expressions with a single variable to simplify the equation. For example, setting the cube roots equal to a variable can reduce the equation to a simpler form, making it easier to solve. After solving, substitute back to find the original variable's value.