Problem 26 - Algebra 2

Question

Simplify each expression. \(\frac{3 t^{2}}{t+2} \cdot \frac{t+2}{t^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is 3.

Step 1: Identify Common Factors

The given expression is \( \frac{3 t^{2}}{t+2} \cdot \frac{t+2}{t^{2}} \). Notice that the \( t+2 \) in the numerator of the second fraction and \( t+2 \) in the denominator of the first fraction are common factors.

Step 2: Cancel Common Factors

Since \( t+2 \) appears in both the numerator and denominator of the multiplied fractions, they can cancel each other out. After canceling \( t+2 \), the expression simplifies to \( \frac{3 t^{2}}{t^{2}} \).

Step 3: Simplify the Remaining Fraction

Now, simplify \( \frac{3 t^{2}}{t^{2}} \). The \( t^{2} \) in the numerator and denominator cancel out, leaving just 3 as the result.

Key Concepts

Common Factors

In algebraic expressions, a common factor refers to any expression or number that appears in both the numerator and denominator of a fraction or within terms being multiplied. Identifying and canceling these common factors is key to simplifying expressions. For example, in the expression \( \frac{3 t^{2}}{t+2} \cdot \frac{t+2}{t^{2}} \), the term \( t+2 \) appears in the numerator of the second fraction and the denominator of the first. Finding these common factors allows us to effectively streamline the expression by removing redundant parts. This process is akin to simplifying numerical fractions by dividing both the numerator and the denominator by their greatest common divisor.

Fraction Multiplication

Multiplying fractions in algebra involves multiplying the numerators together and the denominators together. It's important to pay attention to the structure of the fraction before beginning multiplication, as simplifying first can often make the task easier. In our example \( \frac{3 t^{2}}{t+2} \cdot \frac{t+2}{t^{2}} \), you can multiply directly, but checking for common factors to cancel first saves time and avoids complex calculations later. Also, understanding fraction multiplication helps enhance comprehension of rational expression manipulations, ensuring a more concrete grasp of algebraic principles.

Cancelling Terms

Cancelling is a simplifying step where terms common in both the numerator and the denominator are removed from the expression to make calculations more straightforward. This is permissible because a term divided by itself equals one (\( \frac{a}{a} = 1 \)), and multiplying by one does not change the value of an expression. In the expression \( \frac{3 t^{2}}{t+2} \cdot \frac{t+2}{t^{2}} \), we notice the term \( t+2 \) is present as a common factor, allowing us to cancel it and considerably simplify the expression. This step reduces the expression to \( \frac{3 t^{2}}{t^{2}} \), paving the way for further simplification.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and operations like addition and multiplication. They form the backbone of algebraic manipulations and require understanding specific concepts such as order of operations, combining like terms, and factoring. The process of simplifying expressions like \( \frac{3 t^{2}}{t+2} \cdot \frac{t+2}{t^{2}} \) involves recognizing and employing these basic rules. When handling algebraic expressions, it is useful to break them down into simpler terms, consider each part, and identify strategies such as factoring or cancelling to achieve a more manageable form. The aim is to transform complex expressions into their simplest, most intuitive forms for easier interpretation and solution.