In Exercises 1–38, solve each radical equation.(x - 2)¹/⁴ = (3x - 8)¹/⁴

Verified step by step guidance

1. Step 1: Recognize that both sides of the equation are raised to the power of \( \frac{1}{4} \). To eliminate the radicals, raise both sides of the equation to the power of 4.
2. Step 2: After raising both sides to the power of 4, the equation becomes \( x - 2 = 3x - 8 \).
3. Step 3: Rearrange the equation to isolate the variable \( x \) on one side. Subtract \( x \) from both sides to get \( -2 = 2x - 8 \).
4. Step 4: Add 8 to both sides to further isolate the term with \( x \), resulting in \( 6 = 2x \).
5. Step 5: Divide both sides by 2 to solve for \( x \), giving \( x = 3 \).

Key Concepts

Radical Equations

Radical equations are equations that involve roots, such as square roots or fourth roots. To solve these equations, one typically isolates the radical expression and then raises both sides of the equation to the power that eliminates the root. This process may introduce extraneous solutions, so it's important to check all potential solutions in the original equation.

Isolating Variables

Isolating variables is a fundamental algebraic technique used to solve equations. It involves rearranging the equation to get the variable of interest on one side and all other terms on the opposite side. In the context of radical equations, this often means moving the radical expression to one side before applying operations to eliminate the root.

Extraneous Solutions

Extraneous solutions are solutions that emerge from the process of solving an equation but do not satisfy the original equation. This is particularly common in radical equations, where squaring both sides can introduce solutions that are not valid. Therefore, it is crucial to substitute any found solutions back into the original equation to verify their validity.