Problem 23 - A Graphical Approach to College Algebra

Question

Solve each equation by hand. Do not use a calculator. $$2 x^{1 / 3}-5=1$$

Step-by-Step Solution

Answer

The solution is \(x = 27\).

Step 1: Isolate the Radical

The equation is given as \(2 x^{1/3} - 5 = 1\). The first step is to add 5 to both sides of the equation to isolate the term with the cube root: \[2 x^{1/3} = 1 + 5\]\[2 x^{1/3} = 6\].

Step 2: Solve for the Radical

Next, divide both sides by 2 to solve for \(x^{1/3}\):\[x^{1/3} = \frac{6}{2}\]\[x^{1/3} = 3\].

Step 3: Eliminate the Radical

To eliminate the cube root, raise both sides of the equation to the power of 3:\[x = 3^3\].

Step 4: Calculate the Cube

Compute \(3^3\) to find the value of \(x\):\[x = 27\].

Step 5: Verify the Solution

Substitute \(x = 27\) back into the original equation to verify:\[2 (27)^{1/3} - 5 = 2 \times 3 - 5 = 6 - 5 = 1\].The left-hand side equals the right-hand side, confirming the solution is correct.

Key Concepts

Cube Roots

Understanding cube roots is crucial when solving radical equations. A cube root is a special type of radical symbol denoted by \(x^{1/3}\) or \(\sqrt[3]{x}\). It represents a number which, when multiplied by itself three times, results in the original number. For example, the cube root of 27 is 3 because \(3 \times 3 \times 3 = 27\).

When working with cube roots in equations, you often want to isolate the term containing the cube root. This allows you to eliminate the radical and solve for the variable. In the exercise, once we have the cube root term \(x^{1/3} = 3\), we can eliminate the cube root by raising both sides to the power of three, meaning \((x^{1/3})^3 = x\). This process leaves us with \(x = 3^3\), or simply \(x = 27\).

This step forms a key part of solving equations with cube roots, allowing you to move seamlessly from a more complex radical expression to a straightforward linear equation.

Exponents

Exponents are a fundamental concept in mathematics, often appearing in equations involving radicals like cube roots. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, \(3^3\) stands for \(3 \times 3 \times 3\), resulting in 27.

In the context of solving radical equations, exponents are used to "undo" a radical term. If you have a term such as \(x^{1/3}\), raising it to the third power cancels out the cube root, leaving you with \(x\). This process simplifies solving the equation. Exponents also help in verifying solutions. As shown in the verification step, we compute \(3^3 = 27\) to confirm our answer. Always remember that understanding the relationship between exponents and radicals is critical for solving and understanding such equations.

Isolation of Radicals

Isolation of radicals is often the first and crucial step when solving equations involving radicals such as cube roots. The goal is to have the radical expression alone on one side of the equation so that it can be further manipulated or simplified.

In the given exercise, we started with the equation \(2 x^{1/3} - 5 = 1\). Here, the radical expression is \(x^{1/3}\). By adding 5 to both sides, we move towards isolating the radical term:

• \(2 x^{1/3} = 6\)
• Then, dividing both sides by 2 gives \(x^{1/3} = 3\)

Once isolated, solving the radical by raising it to the power that cancels the root is straightforward. This process demonstrates the importance of isolating the radical, as it simplifies subsequent steps and leads to the eventual solution with ease. Always aim to have your radical standing alone before attempting further operations.