Solve each equation. ∛(4x+3)=∛(2x-1)

Verified step by step guidance

1. Recognize that the equation involves cube roots on both sides: \(\sqrt[3]{4x+3} = \sqrt[3]{2x-1}\).
2. Since the cube root function is one-to-one, set the expressions inside the cube roots equal to each other: \$4x + 3 = 2x - 1$.
3. Solve the resulting linear equation by isolating \(x\): subtract \$2x\( from both sides to get \)4x - 2x + 3 = -1\(, which simplifies to \)2x + 3 = -1$.
4. Continue solving for \(x\) by subtracting 3 from both sides: \$2x = -1 - 3\(, which simplifies to \)2x = -4$.
5. Finally, divide both sides by 2 to isolate \(x\): \(x = \frac{-4}{2}\).

Key Concepts

Cube Roots and Radicals

A cube root of a number is a value that, when multiplied by itself three times, gives the original number. Understanding how to work with cube roots and radicals is essential for solving equations involving ∛ expressions, as it allows you to isolate variables by eliminating the radical.

Equating Radicals

When two cube roots are equal, their radicands (the expressions inside the cube roots) must also be equal. This principle lets you set the expressions inside the cube roots equal to each other, simplifying the equation to a linear or polynomial form that can be solved more easily.

Solving Linear Equations

After equating the radicands, you often get a linear equation in one variable. Solving linear equations involves isolating the variable on one side using inverse operations like addition, subtraction, multiplication, or division to find the solution.