Solve each equation. See Examples 4–6. ∜(3x+1)=1

Verified step by step guidance

1. Recognize that the equation involves a fourth root: $\sqrt[4]{3x+1}=1$.
2. To eliminate the fourth root, raise both sides of the equation to the power of 4: $(\sqrt[4]{3x+1}{)}^4=1^4$.
3. Simplify both sides: $3x+1=1$.
4. Isolate the variable $x$ by subtracting 1 from both sides: $3x=0$.
5. Solve for $x$ by dividing both sides by 3: $x=0$.

Key Concepts

Radical Equations

Radical equations involve variables within a radical symbol, such as square roots or higher roots. To solve these equations, one typically isolates the radical on one side and then raises both sides of the equation to the power corresponding to the root to eliminate the radical. This process may introduce extraneous solutions, so it's essential to check all potential solutions in the original equation.

Inverse Operations

Inverse operations are mathematical processes that reverse the effect of another operation. For example, squaring is the inverse of taking a square root. In the context of solving radical equations, applying inverse operations allows us to isolate the variable and solve for its value. Understanding how to apply these operations correctly is crucial for finding accurate solutions.

Extraneous Solutions

Extraneous solutions are solutions that emerge from the algebraic manipulation of an equation but do not satisfy the original equation. When solving radical equations, it is common to encounter these solutions after squaring both sides or performing similar operations. Therefore, it is important to substitute any found solutions back into the original equation to verify their validity.