Problem 95 - College Algebra Essentials

Question

Factor and simplify each algebraic expression. $$4 x^{-\frac{2}{3}}+8 x^{\frac{1}{3}}$$

Step-by-Step Solution

Answer

The simplified and factored form of the given expression is \(4x^{1/3}(1/x + 2)\).

Step 1: Identify common factors

The given expression is \(4x^{-2/3} + 8x^{1/3}\). A common factor in each term is \(4x^{1/3}\).

Step 2: Factor out common factors

When we factor out \(4x^{1/3}\), the expression can be written as \(4x^{1/3}(x^{-1} + 2)\).

Step 3: Use the rules of exponents

The power rule states that \(x^{-n} = 1/x^n\), so we can rewrite the term in the parentheses as \((1/x + 2)\).

Key Concepts

Common Factors

Understanding the concept of finding common factors is crucial when simplifying algebraic expressions. This is the process of identifying terms that appear in every part of the expression. It's like finding ingredients that are common in every dish of a meal and taking them out of the equation. For instance, looking at the expression 4x^{-2/3} + 8x^{1/3}, the term 4x^{1/3} is present in both parts of the expression, albeit in a different form for the first term. By recognizing this, it becomes possible to 'pull out' this common factor to simplify the entire expression.

Using the distributive property in reverse, also known as factoring out, allows you to condense the expression into a more manageable form. This is akin to packing up the common ingredients into one box, making them easier to handle. This step is the cornerstone for simplification and must always be performed before moving on to other processes, like applying the rules of exponents.

Rules of Exponents

Exponents might seem daunting, but with a clear understanding of their rules, you can manipulate algebraic expressions like a pro. The rules of exponents are like the rules of a game—once you know them, you can play effectively. One key rule is the power rule, which relates to how terms with exponents are multiplied or divided. It states that when you multiply powers with the same base, you add the exponents; when dividing, you subtract them.

When an exponent is negative, such as in x^{-n}, it's equivalent to 1/x^n. This means that you're dealing with the reciprocal of the base raised to the positive exponent. In our expression, x^{-1} is thus rewritten as 1/x. This small switch, based on the rule, can transform a complex-looking expression into something far less intimidating, ready to be simplified further. Keep in mind; these rules are the backbone of algebra and must be mastered for successful expression manipulation.

Simplifying Expressions

Simplifying expressions by combining like terms and using mathematical operations is the finishing touch in algebraic expression manipulation. Consider it as the act of tidying up your mathematics 'workbench' after dealing with the common factors and exponent rules. After factoring out the common factors in our expression, we were left with 4x^{1/3}(x^{-1} + 2). At this point, you assess each term within the parentheses and combine them if possible.

Since one term is 1/x and the other is a constant, they aren't 'like terms' and can't be combined. However, simplifying an expression goes beyond just combining terms; it includes making the expression look 'neater' or more conventional. The end goal is to ensure that anyone reading the expression can understand it easily and see immediately what it signifies. In essence, simplifying makes the magic of mathematics accessible to everyone, not just to those who wielded the wand.