Problem 110 - Intermediate Algebra

Question

a. \(\frac{n+2}{4}-\frac{8}{4 n+8}\) b. \(\frac{n+2}{4} \cdot \frac{8}{4 n+8}\)

Step-by-Step Solution

Answer

a: \(\frac{n^2 + 4n - 4}{4(n+2)}\), b: \(\frac{1}{2}\)

Step 1: Simplify the First Expression

To simplify the expression \(\frac{n+2}{4} - \frac{8}{4n+8}\), we need to find a common denominator. Notice that \(4n + 8\) can be factored as \(4(n + 2)\). Therefore, the expression becomes \(\frac{n+2}{4} - \frac{8}{4(n+2)}\). To combine these, the common denominator would be \(4(n+2)\). This transforms the first term into \(\frac{n+2}{n+2} \frac{n+2}{4} = \frac{(n+2)(n+2)}{4(n+2)}\). Thus, the expression becomes \(\frac{(n+2)^2 - 8}{4(n+2)}\). Now, simplify the numerator: \((n+2)^2 - 8 = n^2 + 4n + 4 - 8 = n^2 + 4n - 4\). The simplified expression is: \(\frac{n^2 + 4n - 4}{4(n+2)}\).

Step 2: Simplify the Second Expression

The second expression is \(\frac{n+2}{4} \cdot \frac{8}{4n+8}\). First, factor \(4n + 8\) as \(4(n + 2)\). The expression becomes \(\frac{n+2}{4} \cdot \frac{8}{4(n+2)}\). The \(n+2\) in the numerator and the \(n+2\) in the denominator cancel out, leading to \(\frac{8}{16} = \frac{1}{2}\). Since there are no variables left, this is the most simplified form of the expression.

Key Concepts

Common Denominator

When working with fractions that have different denominators, you need to find a common denominator to combine them. Having a common denominator means the denominators of the fractions are the same.
This is essential in addition or subtraction of fractions.
To find a common denominator, you either adjust both fractions to have the same denominators or factor common terms.

• In our example, \(4n + 8\) factors into \(4(n + 2)\).
• This simplifies the expression, allowing us to establish \(4(n+2)\) as the common denominator.

By setting both fractions over a common denominator, you can easily subtract or add the numbers on top while keeping the denominator the same.
This simplification step is crucial for making calculations manageable and correct.

Factoring Expressions

Factoring is a method used to simplify expressions by breaking them down into products of simpler expressions, usually to make mathematical operation easier.
In our case, factored form is essential for creating the common denominator.

• The expression \(4n + 8\) can be rewritten as \(4(n+2)\). This is achieved by identifying the greatest common factor, which in this case is 4.

Factoring expressions allows for the cancellation of terms in multiplication or division.
Without factoring, complex algebraic operations can become cumbersome, but with it, they become clearer and simpler.
This process helps in simplifying fractions as well as in finding solutions to equations more efficiently.

Simplifying Expressions

Simplifying expressions involves reducing them to the simplest form, without changing their value.
In algebra, this often means combining like terms, finding a common denominator, and factoring expressions. Simplification makes equations and expressions much easier to work with.

• The expression \(\frac{n+2}{4} - \frac{8}{4(n+2)}\) is simplified by finding the common denominator \(4(n+2)\), enabling us to combine and reduce the fractions.
• Similarly, in multiplication, recognizing the factorial components, such as canceling the \(n+2\) in both the numerator and denominator, results in a streamlined calculation \( \frac{1}{2}\).

Simplification reduces error potential in calculations and helps you see the underlying relations within expressions.
It enhances both understanding and solving ability for algebraic problems.