Solve each equation. See Examples 4–6. ∜(x^2+2x)= ∜3

Verified step by step guidance

1. Recognize that both sides of the equation involve the fourth root, which can be expressed as raising to the power of $\frac{1}{4}$.
2. To eliminate the fourth root, raise both sides of the equation to the power of 4: $(\sqrt[4]{x^2+2x}{)}^4=(\sqrt[4]{3}{)}^4$.
3. Simplify both sides: $x^2+2x=3$.
4. Rearrange the equation to form a standard quadratic equation: $x^2+2x-3=0$.
5. Solve the quadratic equation using the quadratic formula, factoring, or completing the square to find the values of $x$.

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 that eliminates the radical. This process may introduce extraneous solutions, so it's essential to check all potential solutions in the original equation.

Properties of Exponents

Understanding properties of exponents is crucial when manipulating equations involving powers. For instance, when raising a radical to a power, one can apply the property that states (a^m)^n = a^(m*n). This concept helps in simplifying expressions and solving equations by allowing the transformation of roots into fractional exponents, making calculations more manageable.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. They can often be solved using factoring, completing the square, or the quadratic formula. In the context of the given problem, the expression x^2 + 2x can be rearranged into a standard quadratic form, allowing for the application of these methods to find the values of x.