Problem 100 - Algebra and Trigonometry

Question

Factor and simplify each algebraic expression. $$\left(x^{2}+3\right)^{-\frac{1}{3}}+\left(x^{2}+3\right)^{-\frac{1}{3}}$$

Step-by-Step Solution

Answer

The simplified expression is \(2\left(\left(x^{2}+3\right)^{-\frac{1}{3}}\right)\)

Step 1: Identify like terms

In this expression, \(\left(x^{2}+3\right)^{-\frac{1}{3}}\) and \(\left(x^{2}+3\right)^{-\frac{1}{3}}\) are exactly the same, hence they are like terms and can be combined by adding.

Step 2: Simplify the expression

Since the two terms are identical, adding them together results in 2 times that term, \(2\left(\left(x^{2}+3\right)^{-\frac{1}{3}}\right)\)

Step 3: Identify factors for perfect square

Although the resulting term is simplified, it can't be factored further since there isn't any factor that can be assumed as a perfect square to simplify the expression.

Key Concepts

Factorization

Factorization is all about breaking down an expression into simpler elements that, when multiplied together, produce the original expression. Think of it as reverse multiplication. It’s like baking a cake: you're given the cake, and you need to figure out the ingredients involved in making it.
In the exercise, we attempt to factor the algebraic expression \[2\left(x^{2}+3\right)^{-rac{1}{3}} \] However, this expression involves exponents and roots, which makes it a bit trickier. In some cases, expressions can be rewritten as a product of simpler factors. But here, the challenge lies in the non-standard exponent of \(-\frac{1}{3}\). This makes factorization based on standard methods quite complex. Despite this, understanding how to decompose terms helps identify opportunities for simplifications, as we'll soon explore.

Like Terms

In algebra, like terms are terms that contain the same variables raised to the same powers. They only differ in their coefficients. Knowing how to identify and combine like terms is crucial for simplifying expressions.
In the exercise, both terms are \(\left(x^2 + 3\right)^{-\frac{1}{3}}\). Because these terms are alike, they can be combined. They differ only by a factor of 1, making it easy to simplify them into a single term \[2\left(x^2 + 3\right)^{-\frac{1}{3}}\]. Recognizing and merging like terms greatly simplifies the problem and reduces potential errors as you work through more complex algebraic expressions.

Simplification

Simplification is the process of making an expression easier to understand or use. This involves combining like terms, reducing fractions, or applying basic arithmetic operations.
For the expression in the exercise, after identifying the like terms, we combined them to form \[2\left(x^2 + 3\right)^{-\frac{1}{3}}\]. This step eliminates any redundant expressions and reduces it to its simplest form. Sometimes simplification can involve factoring or using other algebraic operations, but here, given the form, simply recognizing and combining is sufficient.
Remember:

• Combine like terms.
• Reduce fractions to simplest forms.
• Use basic arithmetic where possible.

Simplification helps streamline algebraic processes and is a key skill in solving mathematical problems.

Perfect Squares

A perfect square is an expression that is the square of a binomial. Perfect squares take forms that allow further simplification by rewriting them as squared terms. For instance, \((x + 1)^2\) is a perfect square. Unraveling or identifying the possibility of perfect squares is a critical part of simplification.
In this exercise, the term \[2\left(x^{2}+3\right)^{-rac{1}{3}}\] doesn't lend itself directly to being identified or transformed as a perfect square. The exercise confirmed there is no potential factor that classifies as a perfect square here. This means, even though we simplify the terms, certain expressions remain in their simplified but unfactored perfect form. The understanding of perfect squares helps assess simplification opportunities in algebraic expressions but not all scenarios allow transformation into perfect squares.