Problem 44 - Precalculus : Building Concepts and Connections

Question

Solve the radical equation to find all real solutions. Check your solutions. $$\sqrt[3]{5 x-3}=\sqrt[3]{4}$$

Step-by-Step Solution

Answer

The real solution for the equation \( \sqrt[3]{5 x-3} = \sqrt[3]{4} \) is \( x = \frac{7}{5} \).

Step 1: Eliminate the Cube Roots

To eliminate the cube roots, raise both sides of the equation to the power of 3: \((\sqrt[3]{5 x-3})^3 = (\sqrt[3]{4})^3\). This simplifies to \(5x - 3 = 4\).

Step 2: Isolate the Variable

To isolate the variable x, first add 3 to both sides of the equation: \(5x - 3 + 3 = 4 + 3\). This gives \(5x = 7\). Then, divide both sides of the equation by 5 to isolate x: \(x = \frac{7}{5}\).

Step 3: Check the Solution

To check if \(\frac{7}{5}\) is a solution, transform and substitute it back into the original equation: \(\sqrt[3]{5 * \frac{7}{5} -3} = \sqrt[3]{4}\). This gives \(\sqrt[3]{4} = \sqrt[3]{4}\), confirming that \(\frac{7}{5}\) is a solution.

Key Concepts

Cube Roots

Cube roots are special mathematical operations that can help us solve certain equations. They are similar to square roots, but instead of looking for a number that can be squared to get a certain value, we need a number that can be cubed. When you see \( \sqrt[3]{x} \), it means you need to find a number that, when multiplied by itself three times, equals \( x \).

For example, the cube root of 8 is 2, because \( 2 \times 2 \times 2 = 8 \). In equations, understanding cube roots can help us "undo" the operation and solve for variables inside the cube root.

When solving radical equations involving cube roots, the goal is often to eliminate the cube root by raising the expression to the power of three. This technique can make these equations simpler and more straightforward.

Isolating Variables

Isolating variables is an essential technique in solving equations. The idea is to get the variable (often represented as \( x \)) on one side of the equation all by itself. When the variable is isolated, it means you have found the solution.

Let's take the example from the exercise: once you simplify \( 5x - 3 = 4 \), the next step is to move all other terms to the opposite side. Adding 3 to both sides gives you \( 5x = 7 \).

• First, perform operations like addition or subtraction to remove constants or coefficients.
• Then, use division or multiplication to handle any remaining coefficients.

In this case, dividing by 5 isolates \( x \), resulting in \( x = \frac{7}{5} \). Remember, the key to success with isolating variables is to perform the same operation on both sides of the equation to maintain balance.

Checking Solutions

Checking solutions is a vital step to ensure that the process of solving equations was correct. Once you've found a potential solution, it's important to substitute it back into the original equation to verify. This final step makes sure that the value satisfies the equation fully.

In the example given, after determining that \( x = \frac{7}{5} \), you substitute it back into the original equation: \( \sqrt[3]{5 \times \frac{7}{5} - 3} = \sqrt[3]{4} \). Simplifying this confirms that \( \sqrt[3]{4} = \sqrt[3]{4} \).

Here are a couple of tips:

• Always go back to the very original equation when checking.
• Watch out for any forgotten or invisible operations that could lead to errors.

This process validates that the solution is correct and final, which is essential for complete accuracy in solving equations.