Problem 39 - Beginning and Intermediate Algebra

Question

Solve. $$\sqrt[3]{5 x-2}=-3$$

Step-by-Step Solution

Answer

The solution for the given equation \(\sqrt[3]{5x-2}=-3\) can be found by following these steps: Cube both sides of the equation, simplify, and solve for x. The solution is \(x=-5\).

Step 1: Cube both sides of the equation

To get rid of the cube root, we need to cube both sides of the equation. This will give us: \[(\sqrt[3]{5x-2})^3 = (-3)^3\]

Step 2: Simplify the equation

Now, we simplify both sides of the equation. The cube and the cube root on the left side cancel each other, and we are left with: \[5x - 2 = -27\]

Step 3: Solve for x

To solve for x, we need to isolate the variable. Start by adding 2 to both sides of the equation: \[5x = -25\] Then, divide both sides by 5: \[x = -5\] The solution for the given equation is \(x=-5\).

Key Concepts

Cube Roots

The concept of cube roots revolves around finding a number that, when multiplied by itself twice more, results in the original number. For example, the cube root of 27 is 3, because

• 3 × 3 × 3 = 27.

In equations, cube roots are often represented using the radical symbol with an index of three, denoted as \(\sqrt[3]{\ldots}\). For the equation \(\sqrt[3]{a} = b\), cubing both sides can help eliminate the cube root.
This is achieved because

• \((\sqrt[3]{a})^3 = a\),
• \(b^3 = b \times b \times b\).

Understanding cube roots is essential for solving equations that involve them, as seen in our original exercise where cubing both sides helped to simplify and solve the problem.

Equations

Equations are mathematical statements that assert the equality of two expressions, typically involving variables. When solving equations, the goal is to find the value of the variable that makes the equation true. In algebra, equations can come in various forms and complexities, ranging from simple linear equations to more complex polynomial equations.For equations involving cube roots, it's crucial to first eliminate the root to simplify calculations. As demonstrated in our original exercise, cubing both sides of the equation \(\sqrt[3]{5x-2}=-3\) helped in simplifying it to a linear equation, \(5x - 2 = -27\). This reduction step is critical as it allows better focus on isolating the variable.Maintaining balance is important in equations, meaning any operation performed on one side must be done to the other. This principle guides us as we solve and rearrange the equation to find the unknown variable.

Variable Isolation

Variable isolation is the process of manipulating an equation so that the variable in question stands alone on one side. This technique is pivotal when solving for unknowns. The key is to perform inverse operations to move terms around and simplify the equation.In our example, after simplifying to \(5x - 2 = -27\), the next step was to isolate \(x\). This was achieved by:

• Adding 2 to both sides, resulting in \(5x = -25\).
• Then, dividing both sides by 5, giving \(x = -5\).

These actions illustrate the idea of reversing operations to peel away layers around the variable, thereby isolating it. It's crucial to follow the order of operations and proceed step-by-step to ensure accuracy. Understanding variable isolation enhances the ability to solve not just basic, but also complex algebraic equations.