Problem 58 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\sqrt[3]{2 x}=\sqrt[3]{5 x+2}$$

Step-by-Step Solution

Answer

x = -\frac{2}{3}

Step 1 - Isolate the Cube Root

The given equation is \ \ \ \ \(\sqrt[3]{2x} = \sqrt[3]{5x+2}\). It already has the cube roots isolated on both sides.

Step 2 - Remove the Cube Roots

Cube both sides of the equation to eliminate the cube roots: \ \ \((\sqrt[3]{2x})^3 = (\sqrt[3]{5x+2})^3\). This simplifies to \ \ \(2x = 5x + 2\).

Step 3 - Solve for x

Rearrange the equation to isolate \(x\): \ \ \(2x - 5x = 2\), which simplifies to \ \ \(-3x = 2\). Divide both sides by -3: \ \ \(x = -\frac{2}{3}\).

Step 4 - Verify the Solution

Substitute \(x = -\frac{2}{3}\) back into the original equation to verify: \ \ \(\sqrt[3]{2(-\frac{2}{3})} = \sqrt[3]{5(-\frac{2}{3}) + 2}\), which simplifies to \ \ \(\sqrt[3]{-\frac{4}{3}} = \sqrt[3]{-\frac{10}{3} + 2}\) \ \ \(\sqrt[3]{-\frac{4}{3}} = \sqrt[3]{-\frac{4}{3}}\). Both sides are equal, verifying that \(x = -\frac{2}{3}\) is correct.

Key Concepts

Cube Roots

Cube roots are the numbers that, when raised to the power of three, give the original number. For example, the cube root of 8 is 2, because \(2^3 = 8\). Similarly, the cube root of -8 is -2, because \((-2)^3 = -8\). When solving equations involving cube roots, it's crucial to understand that the cube root function can handle both positive and negative values. This property is important because it means that we don't need to worry about negative signs inside the cube root, unlike with square roots.

Isolating Variables

Isolating the variable is a key step in solving any equation. In the given equation, \(\root[3]{2x} = \root[3]{5x + 2}\), the cube roots are already isolated on both sides. If the cube roots were not isolated, you would need to rearrange the equation to isolate them before proceeding. For example, if you had \(\root[3]{2x + 3} = \root[3]{5x + 2}\), you would need to first subtract 3 from both sides to isolate the cube root term: \(\root[3]{2x} = \root[3]{5x - 1}\). Once the cube roots are isolated, it becomes easier to remove them and solve for the variable.

Cubing Both Sides

To eliminate the cube roots, you can cube both sides of the equation. Cubing reverses the effect of the cube root, allowing you to work with simpler expressions. In the original equation, \(\root[3]{2x} = \root[3]{5x + 2}\), cubing both sides gives you \(\root[3]{2x}^3 = \root[3]{5x + 2}^3\), which simplifies to \(2x = 5x + 2\). This is a much simpler equation to solve. Cubing both sides can be especially useful when dealing with complex terms inside the cube roots, as it allows for easier simplification.

Verifying Solutions

Always verify your solutions by substituting them back into the original equation. After solving the equation \(2x = 5x + 2\) and finding that \(x = -\frac{2}{3}\), substitute \(x = -\frac{2}{3}\) back into the original equation to check your work. In this case, \(\root[3]{2(-\frac{2}{3})} = \root[3]{5(-\frac{2}{3}) + 2}\) simplifies to \(\root[3]{-\frac{4}{3}} = \root[3]{-\frac{10}{3} + 2}\), and further to \(\root[3]{-\frac{4}{3}} = \root[3]{-\frac{4}{3}}\). Both sides are equal, verifying that the solution is correct. Verifying solutions ensures that no mistakes were made during the solving process and solidifies your understanding of the problem.