Problem 86 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{a^{2}+1}{a^{2}-1}+\frac{a}{1-a^{2}} $$

Step-by-Step Solution

Answer

\(\frac{a^2+1-a}{a^2-1}\) is the simplified expression.

Step 1: Identify the Structure of the Expression

The expression given is \( \frac{a^{2}+1}{a^{2}-1}+\frac{a}{1-a^{2}} \). To simplify, we need to combine the two fractions into a single fraction. Observing the denominators, \( a^2 - 1 \) and \( 1 - a^2 \), we can notice that \( 1 - a^2 \) is \( -(a^2 - 1) \).

Step 2: Rewrite Denominator of Second Fraction

The second fraction \( \frac{a}{1-a^{2}} \) can be rewritten using the identity \( 1-a^2 = -(a^2-1) \). Therefore, \( \frac{a}{1-a^{2}} = \frac{-a}{a^2-1} \).

Step 3: Combine the Fractions

Now that both fractions have the denominator \( a^2-1 \), we can combine them. This becomes: \[ \frac{a^2+1}{a^2-1} + \frac{-a}{a^2-1} = \frac{a^2+1-a}{a^2-1} \].

Step 4: Simplify the Numerator

The numerator \( a^2 + 1 - a \) cannot be factored further in a way that will simplify with the denominator \( a^2-1 \). Therefore, the expression is already simplified in terms of combining like terms. The final simplified expression is simply: \( \frac{a^2 + 1 - a}{a^2 - 1} \).

Key Concepts

Simplifying Algebraic Expressions

Simplifying expressions in algebra means reducing the expression to its most concise and compact form while maintaining the same value. This involves a combination of combining like terms, reducing fractions, and applying mathematical operations correctly to achieve a simple form. For example, in the given expression \( \frac{a^2+1}{a^2-1}+\frac{a}{1-a^2} \), we can use techniques like identifying common patterns, rewriting terms, and ensuring we have the same denominators to simplify effectively.
In algebra, simplifying helps make expressions easier to work with in solving equations, comparing expressions, and performing further calculations. A critical point in simplifying is ensuring that nothing is lost from the original expression, meaning the value remains the same even though the form looks different. Always double-check your work by substituting values to verify the expressions are equivalent.

Understanding Fractions in Algebra

Fractions in algebra work similarly to basic arithmetic fractions but can involve variables. When working with fractions like \( \frac{a}{b} \), both \( a \) and \( b \) can be algebraic expressions.

Key points about algebraic fractions:

• The numerator (top part) and denominator (bottom part) can contain variables and numbers.
• Simplifying these fractions often involves finding a common denominator or factoring.
• Remember that division by zero is undefined, so check for values that make the denominator zero.

In the original exercise, we handled fractions by rewriting \( \frac{a}{1-a^2} \) as \( \frac{-a}{a^2-1} \) to share a common denominator with the first fraction, facilitating easier addition. Always start by observing if any terms can be canceled or reduced, and ensure all mathematical properties like distributive, associative, and commutative laws are applied correctly.

Role of Denominators in Simplification

Denominators are crucial in algebraic fractions as they determine how easily fractions can be combined and simplified. A denominator's main job is to show the division aspect of a fraction and to help in finding a common basis among multiple fractions.

In the exercise \( \frac{a^2+1}{a^2-1}+\frac{-a}{a^2-1} \), observing the denominators allowed us to rewrite and combine fractions effectively. Here is a brief guide on dealing with denominators:

• Identify common denominators when adding or subtracting fractions: This simplifies operations and helps avoid creating complex expressions.
• Rewrite fractions to get common denominators if they initially differ: This typically involves finding equivalent expressions or using identities to transform fractions.
• Always beware of values that make denominators zero, as they are undefined in algebra.

By recognizing that \( 1-a^2 \) is the negative of \( a^2-1 \), we could achieve a uniform denominator. This understanding is key to simplifying the expression further and seamlessly performing algebraic operations.