Problem 95 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified form of the given algebraic expression is \(4((1/x^3) + 2x^3)\)

Step 1: Identify the common factor

First off, look for the common factor in both terms. It can be observed that 4 is a common factor as both \(4x^{-3}\) and \(8x^3\) can be divided by 4.

Step 2: Factor out the common factor

Now, factor out the common factor from both terms. This can be done by dividing each term by the common factor 4. The expression becomes \(4(x^{-3} + 2x^3)\).

Step 3: Simplification using exponent rules

Remember that \(x^{-3} = 1/x^3\). Substituting this into the equation, we have \(4((1/x^3) + 2x^3)\). The equation cannot be further simplified.

Key Concepts

Factoring

Factoring is an essential skill in algebra that involves breaking down an expression into simpler components or factors. This process helps simplify complex expressions and makes calculations easier. In the provided exercise, the goal is to identify and factor out the greatest common factor (GCF) from the given terms.

• Identify the GCF: Before factoring, determine the greatest common factor of the terms. Here, both terms have 4 as a common factor.
• Factor Out the GCF: Divide each term by the GCF and rewrite the expression. In this exercise, when dividing both terms by 4, the expression becomes: \(4(x^{-3} + 2x^3)\).

When expressions are factored, they become easier to handle in further algebraic operations, such as solving equations or integrating functions.

Exponent Rules

Exponent rules help in manipulating expressions that contain powers or exponents. Understanding these rules is crucial when working with algebraic expressions that involve different powers of the same base.

• Negative Exponents: A negative exponent indicates the reciprocal of the base. For example, \(x^{-3}\) is equivalent to \(\frac{1}{x^3}\). This means we can rewrite \(x^{-3}\) in a form that is easier to use in further calculations.
• Multiplying Exponents: When multiplying like bases, you add the exponents. However, in this problem, the expressions had different powers so simplification beyond applying the negative exponent rule was not performed.

Having a firm grasp of these rules allows you to easily manipulate expressions, like transforming them into simpler or more standardized forms.

Simplification

Simplification involves reducing an algebraic expression to its simplest form. This makes the expression easier to understand and solve. In this exercise, after factoring, simplification is guided by the exponent rules.To simplify the expression further:

• Convert Exponents: The given expression after factoring has \(x^{-3}\), which is rewritten as \(\frac{1}{x^3}\). This doesn’t change the expression structurally but helps in understanding and working with the terms.
• Add Fractions: If possible, simplify fractions or terms within the expression. In this case, \(4((1/x^3) + 2x^3)\) offers no further arithmetic simplification without specific values for \(x\).

Effective simplification helps in working with expressions, especially when preparing for more complex algebraic problems.