Solve each equation. See Examples 4–6. ∛2x=∛(5x+2)

Verified step by step guidance

1. Start by recognizing that both sides of the equation involve cube roots: $\sqrt[3]{2x}=\sqrt[3]{5x+2}$.
2. To eliminate the cube roots, cube both sides of the equation: $(\sqrt[3]{2x}{)}^3=(\sqrt[3]{5x+2}{)}^3$.
3. Simplify both sides: $2x=5x+2$.
4. Rearrange the equation to isolate $x$ on one side: $2x-5x=2$.
5. Combine like terms to solve for $x$: $-3x=2$.

Key Concepts

Cube Roots

The cube root of a number 'a' is a value 'b' such that b³ = a. In the equation ∛2x = ∛(5x + 2), understanding how to manipulate cube roots is essential. This involves recognizing that if two cube roots are equal, then their radicands (the expressions inside the cube roots) must also be equal, leading to the equation 2x = 5x + 2.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate variables. In solving the equation derived from the cube roots, you will need to combine like terms and isolate 'x' to find its value. This skill is fundamental in algebra, allowing for the solution of various types of equations.

Checking Solutions

After finding a potential solution for 'x', it is crucial to check the solution by substituting it back into the original equation. This step ensures that the solution is valid and satisfies the equation. In this case, substituting the value of 'x' back into the original cube root equation will confirm whether the left-hand side equals the right-hand side.