Problem 28 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{(x+10)^{3}}{x+10} $$

Step-by-Step Solution

Answer

Question: Reduce the given rational expression to its lowest terms: $\frac{(x+10)^3}{x+10}$ Answer: $(x+10)^2$

Step 1: Factor out common factor from the numerator

We start by factoring out the common factor from the numerator, which in this case is \((x+10)\). Since the numerator is \((x+10)^3\), it can be written as \((x+10)(x+10)(x+10)\).

Step 2: Cancel out the common factor

Now we can cancel out the common factor between the numerator and the denominator. In this case, the common factor is \((x+10)\). So, we can cancel out one \((x+10)\) from both the numerator and the denominator.

Step 3: Simplify the expression

After canceling out the common factor, we are left with: $$ \frac{(x+10)(x+10)(x+10)}{x+10} \cdot \frac{1}{(x+10)} $$ $$ = \frac{(x+10)(x+10)}{1}\nonumber $$ So, the simplified expression is: $$ (x+10)^2 $$

Key Concepts

Lowest terms

When dealing with rational expressions, one key objective is to reduce these expressions to their lowest terms. This means simplifying them so they can't be made any simpler without changing the value of the expression. Generally, this involves finding common factors in the numerator and denominator and eliminating them.

The initial rational expression in the exercise is \( \frac{(x+10)^3}{x+10} \). Here, the goal is to see if the numerator and the denominator share any similar components that can be "canceled out". Finding common factors is crucial for this process, as it allows us to shed any excess complexity in the expression. In this example, both the numerator and the denominator contain \((x+10)\), which can be removed to simplify the expression.

• Look for common factors in both the numerator and denominator.
• Cancel out those common factors to achieve the lowest term.

Understanding this process is essential as it applies to many areas within mathematics and allows for easier manipulation of expressions in future steps of solving equations.

Simplify expressions

Simplifying expressions is a critical skill in algebra and allows you to make expressions more manageable. By simplifying, you might change how the expression looks, but not its value. This is often achieved by reducing expressions to their lowest terms, as illustrated in our example exercise.

In the given problem, simplifying the rational expression \( \frac{(x+10)^3}{x+10} \) involves more than just canceling out terms. It requires recognizing how the expression can be transformed into a simpler form. After factoring the numerator, \( (x+10)^3 = (x+10)(x+10)(x+10) \), we remove one instance of \( (x+10) \) from both the top and bottom which leaves us with \( (x+10)^2 \).

• Breaking down complex expressions helps in identifying removable terms.
• Transforming expressions into simpler forms aids clearer understanding and further calculations.

This step not only reduces the expression but also helps set the stage for any further algebraic manipulations needed in solving mathematical problems.

Factorization

Factorization is a method used to break down expressions into products of simpler expressions. It simplifies complex expressions and is a fundamental concept in algebra. By factoring, you can easily identify elements that can be simplified or eliminated.

In the exercise, the numerator \((x+10)^3\) is factored as \((x+10)(x+10)(x+10)\). Identifying the \( (x+10) \) term as a common factor is essential here because it is what allows the expression to be simplified to its lowest terms.

• Start by identifying expressions that can be factored completely.
• Use factorization to reveal common terms that enable the reduction of the expression.

Factoring is particularly helpful in more complex algebraic expressions, where recognizing patterns or common factors can simplify operations and help solve equations more efficiently. This skill becomes increasingly important as students progress to more advanced mathematics.