Problem 95 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The simplified and factored form of the given expression \(4 x^{-\frac{1}{3}}+8 x^{\frac{1}{3}}\) is \(4x^{-\frac{1}{3}}(1+2x^{\frac{2}{3}})\).

Step 1: Identify the Common Factor

Identify any common factors in each term of the expression. Here, the common factor is \(4x^{-\frac{1}{3}}\) because both terms contain this.

Step 2: Factor Out the Common Factor

Factor out the common factor from both terms. This results in \[4x^{-\frac{1}{3}}(1+2x^{\frac{2}{3}})\]

Step 3: Simplify the Expression

Simplify the expression where possible. However in this case, it is already in the simplest form, so no further simplification is possible.

Key Concepts

Common Factor

When approaching algebraic expressions, one of the first steps is to identify a common factor for each term. A common factor is a term that divides all parts of the expression without leaving a remainder.
For the expression \(4 x^{-\frac{1}{3}}+8 x^{\frac{1}{3}}\), the term \(4x^{-\frac{1}{3}}\) is the common factor. Here's why:

• Both terms share the factor of 4.
• Each term is expressed in terms of powers of \(x\).
• The smallest exponent, \(-\frac{1}{3}\), is used as the common power for factoring.

This common factor can be pulled out, simplifying the expression into a more manageable form. Recognizing the greatest common factor in algebra helps to streamline the process of simplification significantly.

Exponent Rules

Exponents denote how many times to multiply a number by itself. Understanding how to manipulate and simplify expressions with exponents is crucial.
In our exercise, we encounter exponents like \(x^{-\frac{1}{3}}\) and \(x^{\frac{1}{3}}\). Here are some fundamental rules for handling exponents:

• Add or Subtract Exponents: When multiplying terms with the same base, add the exponents. When dividing, subtract them. For example, \(x^a \cdot x^b = x^{a+b}\) and \(\frac{x^a}{x^b} = x^{a-b}\).
• Negative Exponents: A negative exponent means reciprocal. Hence, \(x^{-n} = \frac{1}{x^n}\).

Applying these rules helps to simplify terms and to factor expressions more efficiently. In our problem, utilizing the smallest exponent as a common factor enabled us to handle the expression effectively.

Simplifying Expressions

Simplifying expressions makes them easier to handle and understand by reducing complexity. Once you've factored out the common factor, it’s important to check if further simplification is possible.
In our exercise, the expression \(4x^{-\frac{1}{3}}(1+2x^{\frac{2}{3}})\) is already in its simplest form.

• Ensure Simple Form: Expressions should not have redundant factors or complicated fractions.
• Reduce Fractions: If there are terms that can be reduced, do so.
• Consider Exponents: Ensure all exponents are reduced to their lowest terms.

While not all expressions can be simplified further, rewriting them using consistent forms and progressively managing factors and powers helps in maintaining clarity and simplicity.