Problem 69 - Intermediate Algebra

Question

Factor the given factor from the expression. $$ x^{1 / 5} ; x^{2 / 5}-3 x^{1 / 5} $$

Step-by-Step Solution

Answer

The expression factors to \(x^{1/5}(x^{1/5} - 3)\).

Step 1: Identify the Common Factor

Examine the terms in the expression. Both terms, \(x^{2/5}\) and \(-3x^{1/5}\), contain the factor \(x^{1/5}\).

Step 2: Factor Out the Common Factor

Extract \(x^{1/5}\) from the expression. This gives us:\[ x^{1/5}(x^{1/5}) - x^{1/5}(3) \]

Step 3: Write the Expression in Factored Form

Upon factoring out \(x^{1/5}\), rewrite the expression as:\[ x^{1/5}(x^{1/5} - 3) \].

Key Concepts

Common Factors

When factoring expressions, one of the first steps is to identify common factors. A common factor is a term that appears in all parts of the expression you're working with. In this context, you would look at each term in your expression to find out what they share in common. For example, if you have two terms that include some power of a variable, you'll need to determine the smallest power that appears in both terms.

In the exercise provided, we have two terms: \(x^{2/5}\) and \(-3x^{1/5}\). The common factor between these terms is \(x^{1/5}\) because it is the smallest power of \(x\) shared between them. Once you've identified the common factor, the next step in the factoring process is to "factor out" this common factor, allowing you to simplify the expression into a more manageable form.

Exponents

Exponents are a shorthand way to express how many times a number, known as the base, is multiplied by itself. Understanding how to work with exponents is crucial when factoring expressions, especially when such expressions include variables with exponents.

In our example, we encounter fractional exponents: \(x^{2/5}\) and \(x^{1/5}\). Whenever you have terms with fractional exponents, remember that these represent roots as well as powers. For instance, \(x^{1/5}\) means the fifth root of \(x\), and this same concept applies as a repeated multiplication across terms.
Working with fractional exponents:

• Add or subtract exponents when multiplying or dividing similar bases.
• Be cautious with negative exponents; they reflect reciprocal actions.

Factoring Techniques

Factoring techniques involve using strategies to simplify expressions into their base components. One primary technique is to factor out the greatest common factor, as seen in this exercise.

First, determine the common factor, which in our case is \(x^{1/5}\). Next, extract this factor from each term in the expression. This action reduces the complexity of the expression by isolating the repeated element.

To demonstrate, after factoring out \(x^{1/5}\) from \(x^{2/5} - 3x^{1/5}\), you rewrite the expression as \(x^{1/5}(x^{1/5} - 3)\). Remember, the goal of factoring is to break down an expression to be more straightforward and highlight its structure. This makes it easier to handle, especially when solving equations.

• Identify the common factor.
• Extract it from each term.
• Rewrite the expression in factored form.