Problem 36 - Intermediate Algebra

Question

Solve each equation. See Example 5. $$ \sqrt[3]{12 m+4}=4 $$

Step-by-Step Solution

Answer

The solution is \( m = 5 \).

Step 1: Understand the Equation

The equation given is \( \sqrt[3]{12m+4} = 4 \). This means that the cube root of the expression \( 12m+4 \) is equal to 4.

Step 2: Eliminate the Cube Root

To eliminate the cube root, we will cube both sides of the equation. Cubing both sides results in \((\sqrt[3]{12m+4})^3 = 4^3\). Therefore, this simplifies to: \(12m + 4 = 64\).

Step 3: Isolate the Variable Term

We want to solve for \(m\), so we need to isolate \(12m\). Subtract 4 from both sides of the equation to get: \(12m + 4 - 4 = 64 - 4\), which simplifies to \(12m = 60\).

Step 4: Solve for m

Divide both sides of the equation by 12 to solve for \(m\). Performing the division, we get \(m = \frac{60}{12}\). Therefore, \(m = 5\).

Key Concepts

Cube Root Equations

When dealing with cube root equations, it’s important to understand that you’re essentially finding the number that, when multiplied by itself three times, gives you the original expression inside the root. In our exercise, we started with the equation \(\sqrt[3]{12m+4} = 4\). Here, the cube root of \(12m+4\) equals 4.
Cube root equations are typically solved by eliminating the cube root to simplify the expression. By cubing both sides of the equation, we remove the cube root. This means transforming \(\sqrt[3]{12m+4}\) into \(12m+4\) by raising both sides to the power of 3.

• Recognizing the cube root allows you to plan the cubing operation.
• Remember that cubing a cube root cancels the root.

Understanding cube root equations is crucial as it lays the foundation for solving more complex algebraic equations related to exponents and radicals.

Solving Equations

To solve an equation means to find the value of the variable that makes the equation true. In our problem, once we eliminated the cube root, we ended up with a simpler equation: \(12m + 4 = 64\). Solving this requires a few methodical steps.
The first objective in solving equations is to get all terms involving the variable on one side and constants on the other. In the example, this involved subtracting 4 from both sides. This process simplified the equation to \(12m = 60\).

• Follow a step-by-step approach to simplify the problem.
• Keep both sides balanced by doing the same operation.

After simplification, solving involves isolating the variable to find its exact value, which we will explore next.

Isolation of Variables

The "isolation of variables" is a fundamental step in solving equations where you strive to get the variable by itself on one side of the equation. This step makes it easier to see the solution directly.
In our exercise \(12m = 60\), isolating the variable 'm' involved dividing both sides by 12. Division helps remove the coefficient of the variable. So, \(m = \frac{60}{12}\), resulting in \(m = 5\).

• Apply inverse operations to obtain the variable alone.
• Understanding isolation helps improve your algebraic manipulation skills.

Mastering this step simplifies solving equations and prepares you for even more complex algebraic challenges.