Problem 89 - Algebra and Trigonometry

Question

Factor the expression completely. Begin by factoring out the lowest power of each common factor. $$ 2 x^{1 / 3}(x-2)^{2 / 3}-5 x^{4 / 3}(x-2)^{-1 / 3} $$

Step-by-Step Solution

Answer

The expression factors to \(- (x^{1/3})(x-2)^{-1/3}(3x + 4)\).

Step 1: Identify the Common Factors

First, observe the expression: \[ 2x^{1/3}(x-2)^{2/3} - 5x^{4/3}(x-2)^{-1/3} \]Identify the common base terms with their lowest exponents. Here, the common base terms are \(x\) and \((x-2)\). The lowest power of \(x\) is \(x^{1/3}\) and for \((x-2)\) it is \((x-2)^{-1/3}\).

Step 2: Factor Out the Common Terms

Factor out the common terms identified in Step 1 from the expression: Factoring \(x^{1/3}\) and \((x-2)^{-1/3}\) gives:\[x^{1/3}(x-2)^{-1/3} \left[ 2(x-2) - 5x^{3/3} \right]\]

Step 3: Simplify Inside the Bracket

Simplify the expression inside the bracket: \[ 2(x-2) - 5x = 2x - 4 - 5x = -3x - 4 \] Thus, the expression becomes:\[ x^{1/3}(x-2)^{-1/3}(-3x - 4) \]

Step 4: Write the Final Factored Expression

Combine the factors to write the expression in a completely factored form:\[ - (x^{1/3})(x-2)^{-1/3}(3x + 4) \]This is the completely factored form of the original expression.

Key Concepts

Common Factors

Identifying common factors is the first step when factoring algebraic expressions. These factors are parts of the expression that are the same across different terms. In the exercise, we look at two parts that are repeated: the terms with base \(x\), and the terms with base \((x-2)\). Each of these can be factored out, making the expression easier to work with.

• Common bases help to simplify expressions by allowing the removal of repetitive components.
• The lowest power of each base simplifies the process as it is shared across terms.

In our example, we have identified \(x^{1/3}\) and \((x-2)^{-1/3}\) as the lowest powered factors. This allows us to pull these out of the expression, setting the stage for further simplification.

Exponents

Exponents tell us how many times a number, called the base, is multiplied by itself. They are crucial in identifying common factors in expressions. In our exercise, the exponents of the terms with bases \(x\) and \((x-2)\) were 1/3, 4/3, 2/3, and -1/3. Understanding exponents helps us know how to factor out and simplify these bases.

• Finding the smallest exponent of a common base helps simplify expressions.
• Exponents help us track how factors are distributed within expressions.

In this task, we've used exponents to know which terms we can group together by common powers, thus facilitating the factoring process.

Simplification

Simplifying an algebraic expression makes it easier to work with and understand. This involves combining like terms and removing any unnecessary components to make a more straightforward equation. In the solution, once the common factors were extracted, the terms inside the brackets \(2(x-2) - 5x\) were simplified.

• Combine like terms such as \(2x\) and \(-5x\).
• Simplification can reduce an expression's length and make it less complex.

The goal is always to reach an expression that is as simple as possible, making calculations easier down the line. Here, simplification turned \(2x - 4 - 5x\) into \(-3x - 4\), producing a manageable expression within factors.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. They represent mathematical relationships in various forms. Being proficient at manipulating these expressions allows for solving equations and understanding algebra deeply. In the provided exercise, we worked with terms involving variables raised to fractional exponents.

• Algebraic expressions can be simplified, factored, and expanded.
• Understanding each part of an expression, such as its terms, coefficients, and bases, helps in performing operations on them.

Learning to manipulate algebraic expressions helps unlock the ability to solve broader mathematical problems. It is a fundamental skill necessary for mastering algebra and moving towards more advanced mathematics.