Problem 23 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{k-\frac{1}{k}}{\frac{k+1}{k}} $$

Step-by-Step Solution

Answer

Question: What is the simplified form of the given complex rational expression: $$\frac{k-\frac{1}{k}}{\frac{k+1}{k}}$$ Answer: The simplified form of the given complex rational expression is $$k-1$$.

Step 1: Identify the Least Common Denominator (LCD)

To find the least common denominator (LCD) for the fractions involved in the expression, we will look at the denominators of each fraction. In this case, we have: $$ k \quad \text{and} \quad k $$ Since both denominators are the same, the LCD will be k.

Step 2: Multiply numerator and denominator by the LCD

Now, we will multiply both the numerator and the denominator of the complex fraction by the LCD. In this case, we will multiply both by k: $$ \frac{k-\frac{1}{k}}{\frac{k+1}{k}} \cdot \frac{k}{k} = \frac{k(k)-1}{(k+1)} $$

Step 3: Distribute and simplify

We will now distribute the k in the numerator and simplify the expression: $$ \frac{k^2-1}{(k+1)} $$

Step 4: Factor and cancel if possible

Next, we will check if the numerator can be factored and cancel any common factors with the denominator. Factoring the numerator, we have: $$ \frac{(k+1)(k-1)}{(k+1)} $$ Now, we can cancel out the common factor (k+1): $$ \frac{k-1}{1} $$ So, the simplified complex rational expression is: $$ k-1 $$

Key Concepts

Least Common Denominator (LCD)

In algebra, dealing with complex rational expressions involves understanding the Least Common Denominator (LCD). The LCD is essentially the smallest expression that all the denominators can divide into without leaving a remainder. In our specific exercise, the denominators are both "k". This makes finding the LCD straightforward as it remains "k". This step is crucial because having a common denominator allows you to manipulate complex fractions more easily. By multiplying the numerator and the denominator of the complex expression by the LCD, you effectively eliminate the denominators, thus simplifying the expression.

Factoring

Factoring is a fundamental skill in algebra, which involves breaking down expressions into products of simpler expressions. Once the original complex rational expression is simplified by multiplying through by the LCD, it often requires further factoring. For example, the expression might simplify to something like \(k^2 - 1\). By recognizing patterns, such as the difference of squares, you can factor \(k^2 - 1\) as \((k + 1)(k - 1)\). Factoring is essential because it allows us to see if any terms can be cancelled out with the denominators, thereby simplifying the expression further. Effective factoring can transform a complex expression into a much simpler one, just like in our example.

Simplification

Once we've factored the expressions, the next step is simplification. Simplification involves reducing expressions to their simplest form. It means identifying and cancelling out common factors in both the numerator and the denominator. In our exercise, after factoring \((k + 1)(k - 1)\) over \((k + 1)\), we notice \((k + 1)\) as a common factor. We can cancel this common factor, thereby simplifying the expression to \(k - 1\). Simplification not only makes expressions easier to manage, but also helps in solving equations or performing further algebraic operations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. These expressions are the building blocks of algebra. In our given problem, we encounter expressions like \(k\), \(k + 1\), and \(k - 1\). Understanding how to work with algebraic expressions, including the application of operations like addition, subtraction, and factoring, is crucial in simplifying them. Having a firm grasp on these expressions allows you to approach problems methodically, first by identifying parts of the expression that can be combined or reduced. Moreover, simplifying algebraic expressions often requires manipulating them according to mathematical rules to reach the most concise form possible, as seen when simplifying a complex rational expression from its original form to \(k - 1\).