Solve each equation. See Example 7. (x^2+24)^1/4 = 3

Verified step by step guidance

1. Start by isolating the expression inside the fourth root. To do this, raise both sides of the equation to the power of 4 to eliminate the fourth root: \((x^2 + 24)^{1/4} = 3\) becomes \(x^2 + 24 = 3^4\).
2. Calculate \(3^4\) to simplify the right side of the equation.
3. Subtract 24 from both sides of the equation to isolate \(x^2\): \(x^2 = 3^4 - 24\).
4. Solve for \(x\) by taking the square root of both sides: \(x = \pm \sqrt{3^4 - 24}\).
5. Simplify the expression under the square root, if possible, to find the values of \(x\).

Key Concepts

Exponents and Radicals

Understanding exponents and radicals is crucial for solving equations involving powers and roots. In this case, the equation involves a fourth root, represented by the exponent of 1/4. This concept helps in rewriting the equation in a more manageable form, allowing for easier manipulation and solution.

Isolating Variables

Isolating variables is a fundamental technique in algebra that involves rearranging an equation to solve for a specific variable. In this problem, you would first raise both sides of the equation to the fourth power to eliminate the radical, which simplifies the equation and allows for further steps to isolate x.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0. After isolating the variable in the given equation, you may end up with a quadratic equation that can be solved using factoring, completing the square, or the quadratic formula. Recognizing this form is essential for finding the solutions to the original equation.