Problem 34 - Algebra 2 and Trigonometry

Question

In \(3-38,\) solve each equation for the variable, check, and write the solution set. $$ -10+\sqrt[4]{n-2}=-8 $$

Step-by-Step Solution

Answer

The solution set is \( \{18\} \).

Step 1: Isolate the radical expression

Begin by isolating the radical expression \( \sqrt[4]{n-2} \) in the equation. To do this, add 10 to both sides of the equation: \[ -10 + \sqrt[4]{n-2} + 10 = -8 + 10 \] which simplifies to \[ \sqrt[4]{n-2} = 2. \]

Step 2: Eliminate the fourth root

Raise both sides of the equation to the fourth power to eliminate the fourth root: \[ (\sqrt[4]{n-2})^4 = 2^4 \] which simplifies to \[ n-2 = 16. \]

Step 3: Solve for n

Add 2 to both sides to solve for \( n \): \[ n-2 + 2 = 16 + 2 \] which simplifies to \[ n = 18. \]

Step 4: Check the solution

Substitute \( n = 18 \) back into the original equation to verify: \[ -10 + \sqrt[4]{18-2} = -8 \] Simplifying inside the root gives \( \sqrt[4]{16} = 2 \). Thus, the equation becomes \[ -10 + 2 = -8 \] which is true.

Step 5: Write the solution set

Since substituting \( n = 18 \) satisfies the original equation, the solution set is \( \{18\} \).

Key Concepts

Isolating the Radical Expression

When solving equations involving radical expressions, the first step is to isolate the radical to make it easier to eliminate. In our example, the expression is \( \sqrt[4]{n-2} \). The goal is to get this expression alone on one side of the equation. Originally, we have \( -10 + \sqrt[4]{n-2} = -8 \). By adding 10 to both sides, we eliminate the constant term on the left:

• This step simplifies the equation to \( \sqrt[4]{n-2} = 2 \).
• Isolation helps in further manipulation where we will aim to eliminate the root altogether.

Understanding this crucial step sets the stage for more complex operations that follow in solving the equation.

Checking Solutions

After finding a potential solution for \( n \), it is vital to verify if it satisfies the original equation. For the equation \( -10 + \sqrt[4]{n-2} = -8 \), substituting our found value \( n = 18 \) back into it checks our work.

• We substitute to get \( -10 + \sqrt[4]{16} = -8 \).
• Since \( \sqrt[4]{16} = 2 \), the equation simplifies to \( -10 + 2 = -8 \), which holds true.

This step not only confirms our solution but also rectifies any mistakes made previously during the calculations. It's a practice that ensures the solution is valid and accurate.

Understanding the Solution Set

The solution set comprises values that satisfy the given equation. After verifying \( n = 18 \) satisfies the original equation, we write the result in a form called a solution set.

• The solution set is typically denoted with curly braces \( \{18\} \).
• In contexts of more complex equations, a solution set can contain multiple values, or none at all, depending on the situation.

Acknowledging what a solution set represents helps in understanding the results of the process and communicates the outcome of the equation clearly.

Understanding the Fourth Root

Radical expressions can involve various roots such as square, cube, or fourth roots. In our example \( \sqrt[4]{n-2} \), we focus on the fourth root.

• The fourth root of a number is a value that, when multiplied by itself four times, equals the original number.
• For instance, \( \sqrt[4]{16} = 2 \) because \( 2 \times 2 \times 2 \times 2 = 16 \).

Understanding the properties of different roots, like the fourth root, aids in solving a variety of radical equations efficiently and accurately. This foundational knowledge is useful in recognizing and working through similar problems.