Solve each equation. (x²+24x)^1/4 = 3

Verified step by step guidance

1. Start with the given equation: \(\left(x^{2} + 24\right)^{\frac{1}{4}} = 3\).
2. To eliminate the fourth root, raise both sides of the equation to the power of 4: \(\left(\left(x^{2} + 24\right)^{\frac{1}{4}}\right)^4 = 3^4\).
3. Simplify the left side by canceling the exponent and the root, resulting in \(x^{2} + 24 = 3^4\).
4. Calculate \$3^4\( (which is \(3 \times 3 \times 3 \times 3\)) and rewrite the equation as \)x^{2} + 24 = 81$.
5. Isolate \(x^{2}\) by subtracting 24 from both sides: \(x^{2} = 81 - 24\), then solve for \(x\) by taking the square root of both sides, remembering to consider both the positive and negative roots.

Key Concepts

Radical Equations

Radical equations involve variables within roots, such as square roots or fourth roots. Solving them typically requires isolating the radical expression and then eliminating the root by raising both sides of the equation to the appropriate power.

Exponents and Powers

Understanding how to manipulate exponents is crucial, especially fractional exponents which represent roots. For example, a power of 1/4 means the fourth root, so raising both sides of the equation to the 4th power removes the root.

Checking for Extraneous Solutions

When solving radical equations, raising both sides to a power can introduce extraneous solutions that do not satisfy the original equation. It is important to substitute solutions back into the original equation to verify their validity.