Problem 30 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{(x-8)^{2}(x+6)^{4}}{(x-8)(x+6)} $$

Step-by-Step Solution

Answer

Question: Reduce the following rational expression to its lowest terms: $$\frac{(x-8)^{2}(x+6)^{4}}{(x-8)(x+6)}$$ Answer: The reduced rational expression is $$\frac{(x-8)(x+6)^3}{1}$$

Step 1: Identify common factors

The given expression is: $$ \frac{(x-8)^{2}(x+6)^{4}}{(x-8)(x+6)} $$ We can see that \((x-8)\) and \((x+6)\) are common factors in both the numerator and the denominator.

Step 2: Cancel out common factors

Now, cancel out each of these factors by dividing both the numerator and the denominator by the common factors: $$ \frac{(x-8)^{2}(x+6)^{4}}{(x-8)(x+6)} \div \frac{(x-8)(x+6)}{(x-8)(x+6)} $$

Step 3: Simplify the expression

Simplify the expression by subtracting the exponents of the common factors: $$ \frac{(x-8)^{2-1}(x+6)^{4-1}}{1} = \frac{(x-8)(x+6)^3}{1} $$ Since we can't simplify it further, the reduced rational expression is: $$ \frac{(x-8)(x+6)^3}{1} $$

Key Concepts

Factoring

Factoring is an essential skill in algebra, especially when dealing with rational expressions. When we factor an expression, we are breaking it down into simpler components or "factors" that, when multiplied together, give us the original expression. In the context of rational expressions, factoring is crucial for identifying common terms in the numerator and denominator.

For example, consider the expression \( (x-8)(x+6) \). Both \( x-8 \) and \( x+6 \) are factors because they are multiplied together. If we have \( (x-8)^2(x+6)^4 \), it represents \( (x-8) \) multiplied by itself, twice, and \( (x+6) \) multiplied by itself, four times.

When simplifying, our goal is to factor expressions fully so we can cancel out any like terms in the numerator and denominator. This helps us reach the simplest form of the expression. Once you've identified all factorable parts, simplifying becomes much easier.

Simplifying Fractions

Simplifying fractions is an important step in working with rational expressions. It helps reduce the expression to its lowest terms, making it easier to understand and work with.

• First, identify the common factors in both the numerator and the denominator of the fraction.
• Once identified, you can "cancel" or divide these common factors out of the fraction.

In our example, the rational expression starts as \[ \frac{(x-8)^2 (x+6)^4}{(x-8)(x+6)} \] By recognizing that \( (x-8) \) and \( (x+6) \) appear in both the numerator and the denominator, you can divide each by these common factors.

After canceling out the common factors, you're left with \[ \frac{(x-8)(x+6)^3}{1} \], which is the simplest form of the expression. Simplifying fractions in this way helps to make complex expressions much more manageable.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operation signs. They form the backbone of algebra and are crucial in expressing relationships in mathematical form. Simplifying these expressions is often the goal, either by combining like terms or reducing to a simpler form.

• An algebraic expression may consist of multiple terms, such as \( (x-8)^2 \) or \( (x+6)^4 \).
• To simplify these expressions, mathematicians often rely on factoring and the properties of exponents.

In our context, rational expressions are a specific type of algebraic expression – these are fractions where the numerator and the denominator are both polynomial expressions. The essential skill is understanding how to manipulate these using algebra rules, especially with factoring and cancellation, to achieve a simplified expression.Algebraic expressions require a good grasp of rules such as distributive property, factoring, and exponent rules. Once these rules are in place, and understanding is developed, working through complex rational expressions becomes much less daunting, leading to simplified solutions like \[ \frac{(x-8)(x+6)^3}{1} \].