Problem 23 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(\frac{(x+1)^{2}}{x+2} \cdot \frac{x+2}{x+1}\)

Step-by-Step Solution

Answer

The simplified expression is \( x+1 \).

Step 1: Multiply the Fractions

To multiply the fractions, combine the numerators together and the denominators together. This results in \( \frac{(x+1)^2}{x+2} \times \frac{x+2}{x+1} = \frac{(x+1)^2 \cdot (x+2)}{(x+2) \cdot (x+1)} \).

Step 2: Cancel Common Factors

Observe that both the numerator and the denominator have common factors \((x+2)\) and \((x+1)\). Cancel these common factors. As a result, \( \frac{(x+1)^2 \cdot (x+2)}{(x+2) \cdot (x+1)} = \frac{x+1}{1} \).

Step 3: Simplify the Expression

After canceling the common factors, simplify the expression. Since \( \frac{x+1}{1} = x+1 \), the simplified form of the expression is \( x+1 \).

Key Concepts

Understanding Fractions

Fractions are a way to express parts of a whole. A fraction consists of a numerator (the top part) and a denominator (the bottom part).
When working with algebraic fractions, things get a bit more complex, but the basic idea remains the same.
In algebra, fractions can show relationships between expressions. For example, in the exercise, \( \frac{(x+1)^2}{x+2} \) is an algebraic fraction where \((x+1)^2\) is the numerator and \(x+2\) is the denominator.

• The numerator indicates how many parts we have.
• The denominator indicates into how many parts the whole is divided.

To correctly multiply or simplify these fractions, careful handling of both numerators and denominators is essential.

Multiplying Expressions

Multiplying algebraic expressions is similar to multiplying regular numbers, but it involves variables and possibly more steps. When you multiply expressions like fractions, the rule is straightforward.

• Multiply the numerators together to get the new numerator.
• Multiply the denominators together to get the new denominator.

For example, if you have \( \frac{(x+1)^2}{x+2} \cdot \frac{x+2}{x+1} \):
You multiply \((x+1)^2 \cdot (x+2)\) for the new numerator and \((x+2) \cdot (x+1)\) for the new denominator, leading to \( \frac{(x+1)^2 \cdot (x+2)}{(x+2) \cdot (x+1)} \).
Understanding these steps helps prevent errors and simplifies the process of working with more complicated algebraic fractions.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process often includes canceling common factors and combining like terms. Especially with fractions, simplification helps in reducing complexity.
In the exercise solution, simplifying comes after multiplying the expressions. Once you have \( \frac{(x+1)^2 \cdot (x+2)}{(x+2) \cdot (x+1)} \), observe that both the numerator and denominator share common factors: \((x+2)\) and \((x+1)\).

• Cancelling these factors means you divide both the numerator and denominator by these common parts to make the expression simpler.
• When all common factors are canceled, the expression \( \frac{x+1}{1} \) is simplified to \(x+1\).

This process of canceling and simplifying is essential in algebra to understand and solve expressions efficiently.