Problem 30 - Algebra 2 and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution set is \( \{8\} \).

Step 1: Isolate the Variable

To solve for the variable, we begin by cubing both sides of the equation. This is because we are dealing with a cube root, and squaring both sides will remove the root. The equation is \[ \sqrt[3]{x} = 2 \] Cubing both sides gives: \[ (\sqrt[3]{x})^3 = 2^3 \] This simplifies to: \[ x = 8 \]

Step 2: Check the Solution

Before concluding, it’s important to verify that the solution satisfies the original equation. Substitute \( x = 8 \) back into the original equation:\[ \sqrt[3]{8} = 2 \] Calculating the cube root of 8 gives: \[ 2 = 2 \] Since this statement holds true, our solution is verified.

Step 3: Write the Solution Set

Since \( x = 8 \) satisfies the original equation, we express the solution as a set. The solution set for the equation is written as:\[ \{8\} \]

Key Concepts

Solution Set

A solution set is a collection of all possible values that satisfy a given equation. In a mathematical problem, particularly regarding cube root equations like \( \sqrt[3]{x} = 2 \), finding the solution set means identifying which value(s) for \( x \) make the equation true. In our exercise, after solving the equation \( \sqrt[3]{x} = 2 \), we determined that \( x = 8 \). When we express this result, it’s important to denote it using the standard mathematical notation for solution sets. As such, the solution is provided as a set: \( \{8\} \). Why use a set?

• Sets clearly communicate that \( 8 \) is the only solution.
• They are a concise way to represent the answer.
• They capture the concept that solutions are part of a larger mathematical framework of set theory.

When solving equations, always check to ensure that every potential solution is included in your solution set.

Verifying Solutions

Verifying a solution is a crucial step in solving equations that ensures your answer is correct. For cube root equations, this means substituting the found value back into the original equation to check if both sides are equal.Why verify?

• Verification provides confidence that the solution is valid.
• It helps detect any computational errors during the solving process.

In our example, after isolating the variable and solving, we found \( x = 8 \). To verify, we substituTe this back into the equation:\[ \sqrt[3]{8} = 2 \]When calculating the cube root of 8, we find that it's indeed 2, confirming our solution as correct. This step reassures us that the solution satisfies the equation under given conditions.

Isolating Variables

Isolating variables is a fundamental step in equation solving, especially when it comes to cube roots. The goal is to "free" the variable \( x \) so we can find its value without other numbers or mathematical functions obscuring it.How is it done in cube root equations?
In the given equation \( \sqrt[3]{x} = 2 \), isolating \( x \) involves eliminating the cube root. Here’s how it’s typically approached:

• Find the inverse operation: Since \( \sqrt[3]{x} \) involves a cube root, its inverse is to cube.
• Apply the inverse operation: Cube both sides of the equation: \( (\sqrt[3]{x})^3 = 2^3 \).
• Result: The cube and cube root cancel, leaving us with \( x = 8 \).

Understanding how to isolate variables enables you to tackle more complex equations and operations efficiently. Moreover, practicing this discipline ensures a systematic approach to problem-solving in algebraic expressions.