91Ó°ÊÓ

When dealing with fractions in algebra, multiplying fractions is straightforward. To multiply two fractions, you multiply the numerators, then multiply the denominators, and you are left with a new fraction. Let's break it down step-by-step for clarity.

• **Numerators**: First, focus on the top parts of the fractions. Multiply them together to get the numerator of the result.
• **Denominators**: Second, multiply the bottom parts of the fractions to get the new denominator.

The original exercise involves multiplying two fractions: \(\frac{y-1}{y^{2}+1} \cdot \frac{y+1}{y^{2}-1}\).- **Numerator Calculation**: Multiply \( (y-1) \) by \( (y+1) \).- **Denominator Calculation**: Multiply \( (y^2+1) \) by \( (y^2-1) \).This method allows us to handle fractions efficiently, simplifying multiplication into a manageable process.

The difference of squares is a specific type of algebraic expression. It appears as the difference between two squared values, and it can be factored using a simple formula: \( a^2 - b^2 = (a-b)(a+b) \).Let's see how this principle is part of the original problem:

• The expression \((y^2 - 1)\) in the numerator is a difference of squares because it can be rewritten as \((y^2 - 1^2)\), which factors into \((y-1)(y+1)\).
• The expression \((y^2 + 1)(y^2 - 1)\) in the denominator can also be treated using the difference of squares. It simplifies to: \((y^2)^2 - (1)^2 = (y^4 - 1)\).

By identifying and applying the difference of squares, we simplify our work and can rewrite expressions more quickly. It's a useful trick for factoring and simplifying algebraic fractions.

The distributive property is a key tool in algebra that allows you to multiply a sum by distributing the multiplication over each term in the sum. The property is often written as: \(a(b + c) = ab + ac\).In the exercise problem, the distributive property helps expand and simplify terms during multiplication.- When multiplying \((y-1)(y+1)\), apply the distributive property: 1. Distribute \(y\) to \(y+1\): - \(y \cdot y + y \cdot 1 = y^2 + y\). 2. Distribute \(-1\) to \(y+1\): - \(-1 \cdot y + -1 \cdot 1 = -y - 1\).When these products are combined, the middle terms \(y\) and \(-y\) cancel each other out, simplifying to \(y^2 - 1\).Using the distributive property makes it easier to break down and simplify several factors in a multiplication problem, allowing algebraic fraction multiplication to be less overwhelming.