Problem 96 - College Algebra Essentials

Question

Factor and simplify each algebraic expression. $$12 x^{-\frac{3}{4}}+6 x^{\frac{1}{4}}$$

Step-by-Step Solution

Answer

After factoring and simplifying the algebraic expression, it simplifies to \(6 x^{\frac{1}{4}}\).

Step 1: Recognize Common Factors

First, look for any common factors in the given terms of the expression. In this case, the common factors are \(12\) (a number) and \(x^{\frac{1}{4}}\) (a variable term with less power). We can write each term in the expression with these factors in front: \(12 * x^{-\frac{3}{4}} = 12 * x^{\frac{1}{4}} * x^{-1}\) and \(6 * x^{\frac{1}{4}}\).

Step 2: Factor the Expression

After recognizing the common factors in the terms, the next step is to factor the expression using these common factors. Our equation becomes \(12 * x^{\frac{1}{4}} * (x^{-1} + \frac{1}{2})\).

Step 3: Simplify the Expression

The final step is to simplify the expression by evaluating the expression in the parenthesis. This results in \(12 * x^{\frac{1}{4}} * (\frac{1}{2}) = 6*x^{\frac{1}{4}}\).

Key Concepts

Factoring

In algebra, factoring is the process of breaking down expressions into products of simpler expressions. Think of it like finding what you need to multiply together to get the original expression. For example, when you see the expression \(12x^{-\frac{3}{4}} + 6x^{\frac{1}{4}}\), you first look for the greatest common factor. Here, we can pull out the number 6 and the term with the smallest exponent, \(x^{\frac{1}{4}}\). This makes factoring easier and it's a handy tool when simplifying algebraic expressions.

• Identify parts of the expression that are common.
• Factor them out, turning the expression into a product.

Factoring can simplify expressions, making them much easier to work with in future calculations or problem-solving.

Simplifying Expressions

Simplifying algebraic expressions is all about making them easier to read and work with, by reducing them to their simplest form. Once you've factored an expression, it's crucial to also simplify it. For the expression \(12x^{-\frac{3}{4}} + 6x^{\frac{1}{4}}\), after factoring, we end up with the expression in a simpler format. This means combining like terms or evaluating expressions within parentheses.
To simplify:

• Follow any arithmetic operations like addition, subtraction, multiplication, or division as necessary.
• Combine terms wherever possible, especially when they share the same variables and powers.

This will yield a cleaner, more concise expression that represents the same quantity.

Exponents

Exponents tell you how many times to multiply a number by itself. They play a key role in algebraic expressions as they indicate the power of a number or variable. In this problem, we have terms such as \(x^{-\frac{3}{4}}\) and \(x^{\frac{1}{4}}\). Here, exponents are fractions, which can sometimes make problems seem complex but actually are straightforward.
Understand and operate exponents by:

• Remembering that a negative exponent indicates reciprocal.
• Fractional exponents suggest root operations, like a square or cube root.

By dealing comfortably with exponents, you can simplify expressions and equations more effectively.

Common Factors

Finding common factors is like identifying the shared traits between terms in an expression. They are the attributes of an expression that can be factored out entirely, helping reduce complexity. In the example \(12x^{-\frac{3}{4}} + 6x^{\frac{1}{4}}\), the common factors \(12\) and \(x^{\frac{1}{4}}\) are identified immediately. Recognizing these factors lets you see what's shared between terms.

• Look for whole numbers or variables that appear in each term.
• Factor these out, paving the way for simplification.

Efficiently pinpointing common factors refines expressions, making algebraic work simpler and quicker.