Problem 48 - Algebra A Combined Function

Question

To find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of 65 and 81 is found by simplifying \(\frac{65+81}{2} .\) This simplifies to \(\frac{146}{2}=73 .\) Write the average of \(\frac{3}{n}\) and \(\frac{5}{n^{2}}\) as a simplified rational expression.

Step-by-Step Solution

Answer

The average of \(\frac{3}{n}\) and \(\frac{5}{n^2}\) is \(\frac{3n+5}{2n^2}\).

Step 1: Understand the problem

We need to find the average of two expressions, \( \frac{3}{n} \) and \( \frac{5}{n^2} \). To find the average, we sum the two expressions and divide by 2.

Step 2: Sum the expressions

The sum of \( \frac{3}{n} \) and \( \frac{5}{n^2} \) can be found by finding a common denominator, which in this case is \( n^2 \). Rewrite \( \frac{3}{n} \) as \( \frac{3n}{n^2} \). Now, add the fractions: \( \frac{3n}{n^2} + \frac{5}{n^2} = \frac{3n+5}{n^2} \).

Step 3: Divide the sum by 2

To find the average, divide the sum of the fractions by 2. So, we have: \( \frac{3n+5}{n^2} \times \frac{1}{2} = \frac{3n+5}{2n^2} \).

Step 4: Simplify the expression

Ensure the expression is simplified. In this case, \( \frac{3n+5}{2n^2} \) is already in its simplest form since there are no common factors to cancel out.

Key Concepts

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are crucial in algebra because they enable us to perform arithmetic operations with algebraic expressions.
When dealing with rational expressions, it is important to remember that the denominator cannot be zero, since division by zero is undefined.
For example, in our problem, we have two rational expressions: \( \frac{3}{n} \) and \( \frac{5}{n^2} \). Here, both the numerators, 3 and 5, and the denominators, \(n\) and \(n^2\), are polynomials. Because these expressions involve variables in the denominator, it is crucial to identify values for the variables that do not make the denominator zero.

• Keep an eye on the denominators.
• Avoid values that make any expressions undefined.
• Simplify expressions whenever possible for clarity.

Simplifying Fractions

Simplifying fractions involves reducing them to their most basic form, where the numerator and the denominator share no common factors other than 1.
In this problem, we find the sum of the expressions \( \frac{3}{n} \) and \( \frac{5}{n^2} \) by rewriting them with a common denominator and then dividing by 2 to find the average.
Once we combine and simplify, we get \( \frac{3n+5}{2n^2} \). This expression is already simplified because there are no common factors between the numerator \(3n + 5\) and the denominator \(2n^2\).
Remember:

• Always look for common factors to simplify.
• Write each term as much as you can in factored form to easily spot common factors.
• A simplified fraction often provides clearer insights on the properties and dependent variables of the expression.

Common Denominators

Finding a common denominator is essential for adding or subtracting fractions, especially with rational expressions. It allows all terms to have the same denominator, making it possible to directly add or subtract the numerators.
In the exercise, to add \( \frac{3}{n} \) and \( \frac{5}{n^2} \), a common denominator is needed. The least common denominator (LCD) for these fractions is \(n^2\).
Here's why:

• The denominator \(n^2\) is the smallest common multiple of the individual denominators \(n\) and \(n^2\).
• By expressing \(\frac{3}{n}\) as \(\frac{3n}{n^2}\), both fractions now have \(n^2\) as their denominator.
• This method allows for straightforward addition: \(\frac{3n}{n^2} + \frac{5}{n^2} = \frac{3n+5}{n^2}\).

Using a common denominator not only simplifies the arithmetic process but also ensures that the resulting expression is in a form where further operations can be performed effortlessly.