Problem 4 - Elementary and Intermediate Algebra

Question

Fill in the blanks. \(\frac{16 n}{8 n^{3}}\) and \(\frac{2}{n^{2}}\) are _________ expressions. They have the same value for all values of \(n,\) except for \(n=0\)

Step-by-Step Solution

Answer

Equivalent expressions

Step 1: Simplify the First Expression

Let's start by simplifying the expression \( \frac{16n}{8n^3} \). We notice that there is a common factor of \( n \) in both the numerator and the denominator. First, divide both parts by 8:\[ \frac{16n}{8n^3} = \frac{16}{8} \times \frac{n}{n^3} = 2 \times \frac{n}{n^3} = 2 \times \frac{1}{n^2} \]This simplifies to \( \frac{2}{n^2} \).

Step 2: Compare with the Given Expression

Now, compare the simplified expression \( \frac{2}{n^2} \) with the second expression given in the exercise, which is also \( \frac{2}{n^2} \).

Step 3: Determine the Relationship

Since both expressions are \( \frac{2}{n^2} \), and they result in the same value for any variable \( n \) except for \( n=0 \) (because division by zero is undefined), we can conclude that these are equivalent expressions.

Key Concepts

Understanding Algebraic Simplification

Algebraic simplification aims to make expressions easier to work with by minimizing their components and making calculations more straightforward.
In our example, we start with the fraction \( \frac{16n}{8n^3} \). To simplify this, we can break down both the numerator and the denominator:

• Step 1: Identify the common factors. Here, both parts of the fraction have a factor of 8 and \( n \).
• Step 2: Simplify the numeric part: \( \frac{16}{8} = 2 \).
• Step 3: Simplify the algebraic part: \( \frac{n}{n^3} = \frac{1}{n^2} \), because \( n^3 = n \times n^2 \).

Therefore, the fraction simplifies to \( \frac{2}{n^2} \). This process shows how complex algebraic fractions can be broken down into much simpler terms by identifying and removing common factors.

Rational Expressions Explained

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are similar to numerical fractions, but instead, they use variables.
To better understand, let's recall the original expression \( \frac{16n}{8n^3} \), which is a rational expression since it is a fraction composed of polynomial expressions in terms of \( n \).Simplifying rational expressions involves reducing them to their most manageable form, similar to simplifying standard fractions. Here, we obtained the expression \( \frac{2}{n^2} \), which is now more straightforward and equivalent to the original expression except for specific values of \( n \).
This equivalence is significant because the simplified form makes further calculation and comparison easier. Rational expressions help us model real-world scenarios in fields like engineering and economics, where systems can be described by polynomial relationships.

Understanding Undefined Values in Expressions

While working with expressions, particularly rational ones, you may encounter undefined values. These occur when there is division by zero in the expression.
In our case, both \( \frac{16n}{8n^3} \) and its simplified form \( \frac{2}{n^2} \) become undefined when \( n = 0 \). This is because substituting 0 for \( n \) results in a division by zero, which is mathematically undefined.To handle rational expressions effectively:

• Identify variables that cause division by zero. This step is crucial in determining the domain of the expression.
• The domain excludes any values that make the denominator zero. Here, \( n eq 0 \) is necessary for the expression to remain valid.

By recognizing undefined values, you ensure that any calculations or models you create are mathematically sound and reflective of realistic constraints.