Problem 23 - A Graphical Approach to Precalculus with Limits

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 Term with the Variable

Start with the equation \(2x^{1/3} - 5 = 1\). First, move \(-5\) to the other side by adding \(5\) to both sides of the equation, yielding \(2x^{1/3} = 6\).

Step 2: Solve for the Variable Raised to the Power

Now, divide both sides by \(2\) to solve for \(x^{1/3}\). This results in \(x^{1/3} = 3\).

Step 3: Eliminate the Fractional Exponent

To remove the cube root (fractional exponent), raise both sides of the equation \(x^{1/3} = 3\) to the power of \(3\). This gives \(x = 3^3\), or \(x = 27\).

Key Concepts

Isolating Variables

In many algebraic equations, the first step is to isolate the variable, helping to simplify the expression. This means you need to "get the variable by itself." Let's see how this works in the example: \(2x^{1/3} - 5 = 1\). You'll start by moving all other terms to the opposite side of the equation.
Here's how you can isolate:

• Add \(5\) to both sides to counteract the \(-5\) on the left. This simplifies our equation to \(2x^{1/3} = 6\).
• Next, divide both sides by \(2\) so that we have \(x^{1/3}\) on its own. This step results in \(x^{1/3} = 3\).

When isolating variables, remember that whatever you do to one side, you must do to the other. This keeps the equation balanced, like a scale. The goal is always to simplify until the variable stands alone on one side of the equation.

Fractional Exponents

Fractional exponents might seem complex at first, but they are just another way to represent roots. For instance, the equation \(x^{1/3} = 3\) uses a fractional exponent, \(1/3\), which means the cube root of \(x\).

Understanding fractional exponents:

• The numerator of the fraction indicates the power to which the base is raised.
• The denominator indicates the root to be calculated. Here, \(x^{1/3}\) implies the cube root of \(x\).
• To eliminate the fractional exponent, raise the expression to the power of its denominator. In this problem, we raise both sides of \(x^{1/3} = 3\) to the power of \(3\) to get \(x = 3^3\).

It's helpful to remember that fractional exponents simplify various mathematical properties but behave according to the same laws as whole number exponents. Practicing these helps build intuition for more complex equations.

Step-by-Step Solutions

Taking systematic steps while solving equations not only avoids mistakes but also makes the process clearer. Step-by-step solutions guide you through the problem systematically.
Why use step-by-step methods?

• You can tackle one part of the problem at a time, ensuring every aspect is covered before proceeding.
• It helps in identifying which operations to use and the correct order to apply them.
• A methodic approach provides a clear path from start to finish, allowing you to understand how the solution evolves.

In our exercise, we demonstrated how each step builds upon the previous one—from isolating the variable, dealing with the fractional exponent, to ultimately finding that \(x = 27\). This unravels the problem in manageable chunks and helps you focus on mastering methods rather than just answers. The beauty of a step-by-step solution is that it enhances learning by emphasizing the "how" and "why" of each operation.