Problem 94 - College Algebra

Question

Factor and simplify each algebraic expression. $$ x^{\frac{3}{4}}-x^{\frac{1}{4}} $$

Step-by-Step Solution

Answer

The factored and simplified form of the algebraic expression \(x^{\frac{3}{4}} - x^{\frac{1}{4}}\) is \(x^{\frac{1}{4}}(x^{\frac{1}{2}} - 1)\).

Step 1: Identify the common factor

In the expression \(x^{\frac{3}{4}} - x^{\frac{1}{4}}\), the common factor is \(x^{\frac{1}{4}}\).

Step 2: Factor out the common factor

If you factor out \(x^{\frac{1}{4}}\) from each term in the expression it will be \(x^{\frac{1}{4}}(x^{\frac{1}{2}} - 1)\). Here we use the property of exponents that says when dividing like bases, subtract the exponents. So \(x^{\frac{3}{4}} ÷ x^{\frac{1}{4}} = x^{\frac{3}{4} - \frac{1}{4}} = x^{\frac{1}{2}}\).

Step 3: Simplify the expression

The expression \(x^{\frac{1}{4}}(x^{\frac{1}{2}} - 1)\) is the simplified version of the original expression, because it can't be simplified further.

Key Concepts

Common Factors

When dealing with algebraic expressions, a common factor refers to a factor that is shared by all terms in an expression. Identifying and factoring out common factors can greatly simplify expressions. Imagine factors as the building blocks of terms that are hidden inside the expression. Consider an expression like \(x^{\frac{3}{4}} - x^{\frac{1}{4}}\). Here, both terms share the factor \(x^{\frac{1}{4}}\). This is because \(x^{\frac{3}{4}}\) can be thought of as \(x^{\frac{1}{4}} \times x^{\frac{1}{2}}\) and must include \(x^{\frac{1}{4}}\) to exist.The process to find a common factor includes:1. **Identify terms that share the same base.** Look for similar structures in each term.2. **Determine the smallest exponent.** This is particularly important with exponents; find the term with the lowest power.3. **Factor it out.** Remove this factor from each term to simplify the expression.This process simplifies the handling and reduces complexity when working with expressions.

Simplification

Simplification is about making an expression more understandable and reducing it to its most basic form. It involves several steps, such as combining like terms and distributing any common factors. Simplification enhances clarity and helps to understand more about the nature of the expression.For example, after factoring the expression \(x^{\frac{3}{4}} - x^{\frac{1}{4}}\), you end up with \(x^{\frac{1}{4}}(x^{\frac{1}{2}} - 1)\). No further simplification is possible here. But the initial simplification involved understanding: - **Factoring out common terms.** Removing shared factors helps in reducing unwieldy expressions to simpler parts.- **Using properties of exponents.** Applying rules like subtracting exponents when dividing can transform and simplify terms.The act of simplification is about taking a step back and considering how the terms work together, aiming for the simplest composition possible.

Properties of Exponents

The properties of exponents are crucial tools that help simplify expressions involving exponents. Understanding these properties is like having a toolbox that helps breakdown long and cumbersome terms into easily manageable units.Some key properties include:

• **Product Rule**: \(a^m \times a^n = a^{m+n}\) — multiply terms with the same base by adding their exponents.
• **Quotient Rule**: \(\frac{a^m}{a^n} = a^{m-n}\) — divide terms with the same base by subtracting exponents, as used in our exercise: \(x^{\frac{3}{4}} - x^{\frac{1}{4}} = x^{\frac{1}{2}}\).
• **Power Rule**: \((a^m)^n = a^{m \times n}\) — raise a power to another power by multiplying the exponents.
• **Zero Exponent**: Any base with an exponent of zero is one: \(a^0 = 1\).

By leveraging these properties, expressions like \(x^{\frac{3}{4}} - x^{\frac{1}{4}}\) were simplified. These rules are essential not just in algebra but also in advanced mathematics, where managing powers is foundational.