Problem 36 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{15 y^{2}}{5 y^{3}+15 y^{2}} $$

Step-by-Step Solution

Answer

The simplified rational expression is \( \frac{3}{y+3} \).

Step 1: Factor the denominator

First, let's look at the denominator, which is \( 5y^3 + 15y^2 \). Notice that both terms in the denominator have a common factor. The greatest common factor is 5y^2. By factoring 5y^2 out, we write the expression as \( 5y^2(y + 3) \).

Step 2: Re-write the expression

After factoring the denominator, the expression becomes \( \frac{15y^2}{5y^2(y+3)} \).

Step 3: Simplify the rational expression

Next, simplify the rational expression by canceling out the common factor in the numerator and the denominator. In this case, \( 15y^2 \) in the numerator and \( 5y^2 \) in the denominator share a common factor of \( 5y^2 \). Divide both by \( 5y^2 \) to get \( \frac{3}{y+3} \).

Step 4: Confirm the simplification

To verify, observe that the expression \( \frac{3}{y+3} \) is simplified correctly since dividing the entire numerator and denominator by their greatest common factor leaves no further simplification.

Key Concepts

Factoring

Factoring is a fundamental concept in algebra that involves breaking down an expression into its prime components or simpler expressions that multiply together to give the original expression. Think of it like breaking numbers into their prime factors, but now we do it for algebraic expressions.
In the context of rational expressions, factoring allows us to simplify expressions and solve equations more easily. Let's consider the denominator of our expression, which is \(5y^3 + 15y^2\). To factor it, we look for a common factor in each term. Here, both terms have \(5y^2\) in common. By factoring \(5y^2\) out, we rewrite the expression as \(5y^2(y + 3)\). This step is crucial because it sets up the opportunity to simplify the expression later down the line by canceling out the common factors between the numerator and denominator.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form where no further simplification is possible. This often means no unnecessary parentheses, reduced fractions, and the most compact expression of terms with like variables. After factoring, our rational expression \(\frac{15y^2}{5y^2(y+3)}\) is ready for simplification. The goal is to cancel out any common factors between the numerator and the denominator. Since both the numerator and the denominator have the factor \(5y^2\), we divide them both by \(5y^2\). This removes unnecessary parts and leaves us with \(\frac{3}{y+3}\). By performing this step, we have successfully minimized the rational expression, making it easier to work with in further mathematical analysis or computations.

Common Factors

Common factors play a pivotal role in simplifying both numeric and algebraic expressions. They are the factors shared by two or more numbers or terms, crucial for simplifying expressions and solving equations. When you are simplifying a rational expression like \(\frac{15y^2}{5y^2(y+3)}\), identifying common factors is essential. It allows you to cancel terms and make the expression simpler. In our expression, \(15y^2\) and \(5y^2(y+3)\) both contain \(5y^2\) as a common factor. By identifying and then canceling this common factor from both the numerator and the denominator, we significantly reduce the complexity of our expression to \(\frac{3}{y+3}\). Recognizing common factors is a skill that develops with practice and is an invaluable tool for anyone working through algebraic expressions, making calculations much more straightforward.