Problem 13 - Elementary and Intermediate Algebra

Question

Multiply, and then simplify, if possible. \(\frac{y+2}{y} \cdot \frac{3}{y^{2}}\)

Step-by-Step Solution

Answer

The final simplified expression is \(\frac{3y + 6}{y^3}\).

Step 1: Multiply the Numerators

To solve the problem, we start by multiplying the numerators of the two fractions together. The numerators are \(y+2\) and \(3\). Thus, we have \((y+2) \cdot 3 = 3(y+2)\). This can be expanded to \(3y + 6\).

Step 2: Multiply the Denominators

Next, multiply the denominators of the two fractions. The denominators are \(y\) and \(y^2\). Thus, we multiply them: \(y \cdot y^2 = y^{1+2} = y^3\).

Step 3: Combine into a Single Fraction

We now combine the products from steps 1 and 2 into a single fraction: \(\frac{3y + 6}{y^3}\).

Step 4: Simplify the Fraction

To simplify the fraction, factor the numerator and cancel common factors with the denominator if possible. The numerator \(3y + 6\) can be factored into \(3(y + 2)\). Now, the fraction is \(\frac{3(y + 2)}{y^3}\). Since \(y^3\) does not have any common factors with \(3(y + 2)\), the fraction is already in its simplest form.

Key Concepts

Multiplying Fractions

When multiplying fractions, you multiply their numerators together and their denominators together. This rule applies to all fractions, whether they contain numbers, variables, or algebraic expressions.
In the exercise given, we started by multiplying the numerators: \((y+2)\) and \(3\) to get \(3(y+2)\) which expands to \(3y + 6\). Then, the denominators \(y\) and \(y^2\) were multiplied to result in \(y^3\).
Remember these key steps:

• First, multiply the numerators to find the new numerator.
• Next, multiply the denominators to find the new denominator.

The product of the numerators and the denominators form a new single fraction. Be sure to simplify afterward.

Factoring Algebraic Expressions

Factoring is breaking down a complex algebraic expression into simpler parts. These simpler parts are called factors. For polynomial expressions, you can factor by finding common factors or using special formulas like factoring trinomials.
In our example, the numerator \(3y + 6\) was factored into \(3(y + 2)\). This was achieved by extracting the greatest common factor, which is \(3\) in this situation.

• Identify common factors among the terms in the expression.
• Extract the common factor and express the remaining terms as a product.

Factoring makes simplification easier, which is critical in simplifying algebraic fractions.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing them to their simplest form. This process often includes factoring and canceling out common factors in the numerator and denominator.
After multiplying, we had \(\frac{3(y + 2)}{y^3}\). Since there are no common factors between \(3(y + 2)\) and \(y^3\), the fraction is in its simplest form. The purpose of simplification is to make expressions easier to understand and work with.

• Factor the numerator and denominator if possible.
• Cancel any common factors that appear in both the numerator and the denominator.

Always ensure that you confirm the fraction cannot be simplified further.