Solve each equation. See Examples 4–6. ∜(x-15)=2

Verified step by step guidance

1. Recognize that the equation involves a fourth root, which can be rewritten using exponents. The fourth root of an expression \(x - 15\) is written as \((x - 15)^{\frac{1}{4}}\).
2. Rewrite the equation \(\sqrt[4]{x - 15} = 2\) as \((x - 15)^{\frac{1}{4}} = 2\).
3. To eliminate the fourth root, raise both sides of the equation to the power of 4, using the property \((a^{m})^{n} = a^{mn}\). This gives \(x - 15 = 2^{4}\).
4. Calculate \$2^{4}\( (which is \)2 \times 2 \times 2 \times 2$) to simplify the right side of the equation.
5. Solve for \(x\) by adding 15 to both sides, resulting in \(x = 2^{4} + 15\).

Key Concepts

Radical Expressions and Roots

Radical expressions involve roots such as square roots, cube roots, and fourth roots (∜). Understanding how to interpret and manipulate these roots is essential for solving equations involving radicals.

Isolating the Variable

To solve an equation, isolate the variable by undoing operations step-by-step. For radical equations, this often means raising both sides of the equation to the power that eliminates the root.

Checking for Extraneous Solutions

When both sides of an equation are raised to a power, extraneous solutions can arise. Always substitute solutions back into the original equation to verify their validity.