Problem 86 - College Algebra

Question

In Exercises 85-94, factor and simplify each algebraic expression. $$x^{3 / 4}-x^{1 / 4}$$

Step-by-Step Solution

Answer

The factored and simplified form of \(x^{3 / 4}-x^{1 / 4}\) is \(x^{1/4}(x^{1/2} - 1)\)

Step 1: Identify the Common Factor

In \(x^{3 / 4}-x^{1 / 4}\), it can be noticed that both terms have \(x^{1 / 4}\) in common. Therefore, this can be factored out.

Step 2: Factor out the Common Term

Factoring out the common term \(x^{1 / 4}\) from \(x^{3 / 4}-x^{1 / 4}\) gives \(x^{1 / 4}(x^{1 / 2} - 1)\). Here, the remaining term \(x^{1 / 2}\) is obtained by subtracting the exponent 1/4 from 3/4, according to the rule \(x^{a}-x^{b}=x^{a-b}\).

Step 3: Simplify the Expression

The expression is already in its simplest form since there are no more common factors to be factored out.

Key Concepts

Common Factor in Algebra

In algebra, identifying a common factor is a key step in simplifying expressions. A common factor is essentially a term that is present in every part of the algebraic expression. In the expression \(x^{3/4} - x^{1/4}\), both terms have \(x^{1/4}\) as a common factor. This means that each term can be divided by \(x^{1/4}\).

• The common factor is the smallest power of a variable or term that divides each term evenly.
• Identifying a common factor helps in breaking down the expression into simpler components.

Once the common factor is identified, it is factored out of the expression. This process of factoring helps to reduce the complexity of solving the expression by dealing with smaller and simpler components instead of the whole at once.

Simplifying Expressions

Simplifying expressions is an important skill in algebra, aimed at making an expression easier to work with by reducing it to its simplest form. When you simplify an expression like \(x^{1/4}(x^{1/2} - 1)\), you're ensuring that there are no further common factors that can be extracted or reduced.

• Check for common factors in each term within the brackets after factoring out any common term.
• Combine like terms where possible.
• Keep expressions as tidy as possible, removing any unnecessary components.

To assess whether an expression is fully simplified, ensure that there are no further options for factoring or reducing coefficients and variable powers.

Exponent Rules in Algebra

Understanding exponent rules is crucial for correctly simplifying expressions and managing algebraic terms with powers. In the expression \(x^{3/4} - x^{1/4}\), exponent rules help us understand how to manipulate and reduce powers when a common factor is present.

• According to exponent rules, when dividing terms with the same base, subtract the exponents: \(x^{a} / x^{b} = x^{a-b}\).
• For multiplication, add exponents: \(x^{a} \cdot x^{b} = x^{a+b}\).
• Apply these rules systematically to simplify complicated expressions.

By factoring \(x^{1/4}\) out of \(x^{3/4} - x^{1/4}\), you effectively use the rule \(x^{a} - x^{b} = x^{a-b}\) to find that the remaining term is \(x^{1/2}\). This reduction helps in clearly showing the expression’s structure and simplifies further calculations.