Solve each equation. ∛2x=∛(5x+2)

Verified step by step guidance

1. Start with the given equation: \(\sqrt[3]{2x} = \sqrt[3]{5x + 2}\).
2. Since both sides are cube roots and the cube root function is one-to-one, set the radicands equal to each other: \$2x = 5x + 2$.
3. Rearrange the equation to isolate the variable terms on one side: \$2x - 5x = 2$.
4. Simplify the left side: \(-3x = 2\).
5. Solve for \(x\) by dividing both sides by \(-3\): \(x = \frac{2}{-3}\).

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 manipulate cube roots and rewrite expressions involving radicals is essential for solving equations like ∛(2x) = ∛(5x + 2).

Equating Radicals

When two cube roots are equal, their radicands (the expressions inside the roots) must be equal, provided both sides are defined. This allows us to set 2x equal to 5x + 2 and solve the resulting algebraic equation.

Solving Linear Equations

After equating the radicands, the problem reduces to solving a linear equation in one variable. This involves isolating the variable on one side using inverse operations like addition, subtraction, multiplication, or division.