Solve each equation. See Example 7. (x²+24)^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 applying the power of a power rule: \(x^{2} + 24 = 3^{4}\).
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

Solving Radical Equations

Radical equations involve variables within roots, such as square roots or fourth roots. To solve them, isolate the radical expression and then raise both sides of the equation to the power that eliminates the root, ensuring to check for extraneous solutions.

Exponents and Powers

Understanding how to manipulate exponents is crucial, especially fractional exponents which represent roots. For example, raising a quantity to the 1/4 power means taking the fourth root, and reversing this requires raising both sides to the 4th power.

Checking for Extraneous Solutions

When solving equations involving even roots, raising both sides to a power can introduce solutions that don't satisfy the original equation. Always substitute solutions back into the original equation to verify their validity.