Solve each equation. ⁵√ 2x = ⁵√ 3x+2

Verified step by step guidance

1. Recognize that the equation involves fifth roots on both sides: \(\sqrt[5]{2x} = \sqrt[5]{3x + 2}\). Since the fifth root function is one-to-one, you can set the radicands equal to each other to solve for \(x\).
2. Set the expressions inside the fifth roots equal: \$2x = 3x + 2$.
3. Rearrange the equation to isolate \(x\) by subtracting \$3x\( from both sides: \)2x - 3x = 2$.
4. Simplify the left side: \(-x = 2\).
5. Solve for \(x\) by multiplying both sides by \(-1\): \(x = -2\).

Key Concepts

Radical Equations

Radical equations involve variables inside a root, such as square roots or fifth roots. To solve them, you often isolate the radical expression and then eliminate the root by raising both sides to the power that corresponds to the root's index.

Properties of nth Roots

The nth root of a number is the value that, when raised to the nth power, gives the original number. For example, the fifth root of a number x is a value y such that y^5 = x. Understanding this helps in rewriting and solving equations involving roots.

Solving Equations by Equating Radicals

When two nth roots are equal, their radicands (the expressions inside the roots) must be equal, provided the roots are defined for those values. This allows you to set the expressions inside the roots equal to each other and solve the resulting equation.