Problem 88 - College Algebra

Question

In Exercises 85-94, factor and simplify each algebraic expression. $$12 x^{-3 / 4}+6 x^{1 / 4}$$

Step-by-Step Solution

Answer

The simplified and factored expression is \(6x^{-3/4}(2+x)\).

Step 1: Identify the Common Term

First, look at the two terms of the expression \(12 x^{-3 / 4}\) and \(6 x^{1 / 4}\). The common term here would be the smallest power of \(x\) and the common number. In this case, the smallest power is \(-3 / 4\) and the common number is 6.

Step 2: Factor the Common Term

Next step is to factor out the common term. So, \(12 x^{-3 / 4}+6 x^{1 / 4}\) becomes \(6x^{-3/4}(2+x^{1})\).

Step 3: Simplify the Expression

The last step is to simplify the factored expression. Applying the rules of exponents to simplify \(x^{1}\) would result in just \(x\). So, \(6x^{-3/4}(2+x^{1})\) simplifies to \(6x^{-3/4}(2+x)\).

Key Concepts

Simplifying Algebraic Expressions

When dealing with algebra, simplifying expressions is a fundamental skill that can help make complex problems more manageable. Simplification often involves reducing the expression to its most basic form by performing operations such as combining like terms, factoring, and using exponent rules. In our textbook example with the expression 12x^{-3/4} + 6x^{1/4}, simplification starts by identifying common factors among the terms.

Once a common term is identified, you can 'factor it out'. This means to express the original terms as a product of the common term and terms that are unique to each summand. In the case of our example, factoring out 6x^{-3/4} results in a simpler expression where the remaining terms inside the parenthesis are easier to manage. Finally, to ensure the expression is as simple as possible, apply exponent rules to combine terms with the same base and manipulate the expression accordingly; in our example, this leaves us with 6x^{-3/4}(2+x), an expression notably simpler than the one we started with.

Common Term Identification

Identifying common terms is like finding the thread that binds elements of an algebraic expression together. It's a crucial step in factoring, as it simplifies the process. In our focus expression, the common terms are identified by looking for shared factors in both coefficients and variables.

The coefficients 12 and 6 have a common factor of 6. When it comes to the variable x, the exponent rules teach us that we can use the term with the smallest exponent when factoring out. Therefore, x^{-3/4} is the common term for the variable. By extracting this commonality, we're setting up the stage for a much more streamlined expression. Remember, the aim is to pull out the most that we can from each term, yielding a simplified product that reflects the intrinsic connection between the terms.

Exponent Rules

Understanding exponent rules is essential when working with algebraic expressions, especially during the factoring and simplifying process. These rules explain how to handle operations involving powers and can drastically streamline expressions. For example, a key rule is that when multiplying terms with the same base, you add their exponents.

In our example, though, we're actually using a different rule: to simplify the factored expression, we recognize that any term raised to the power of one is simply the term itself. Hence, x^{1} = x. Additionally, whenever a term with an exponent is factored out, the exponents in the resulting expression are adjusted accordingly by subtracting. Grasping these rules helps clarify why, after factoring out x^{-3/4}, we're left with the term x standing alone in the parenthesis, transforming x^{1/4} into simply x because 1/4 - (-3/4) = 1.