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Chapter 7: Problem 24

Multiply, and then simplify, if possible. \(\frac{(y-3)^{2}}{y-5} \cdot \frac{y-5}{y-3}\)

Short Answer

Expert verified
The simplified expression is \(y-3\).

Step by step solution

01

Expand the Expression

First, identify the components of the expression. We have \((y-3)^2 / (y-5)\) and \((y-5) / (y-3)\). When multiplied, we get: \[\frac{(y-3)^2}{y-5} \cdot \frac{y-5}{y-3}\]
02

Cancel Common Terms

Notice that \(y-5\) appears in both the numerator of the second fraction and the denominator of the first fraction. Similarly, \(y-3\) appears in \((y-3)^2\) and the denominator of the second fraction. Cancelling these gives us: \[\frac{y-3}{1}\]
03

Simplify the Expression

The expression \(\frac{y-3}{1}\) simplifies to just \(y-3\). This is the simplified result of the multiplication.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Expressions
When dealing with algebraic fractions, simplifying expressions is a crucial step. It helps to reduce complexity and make computations easier.
Simplification involves reducing the fraction to its simplest form. This is done by breaking down the fraction into smaller, more manageable parts. The key is to identify and eliminate any unnecessary or redundant elements.
To simplify an expression:
  • Identify common factors in the numerator and the denominator.
  • Cancel these common factors whenever possible to reduce the fraction.
  • Ensure that the remaining terms are presented in the simplest form.
By simplifying expressions, calculations become more straightforward and less prone to errors. This process is fundamental when working with polynomials and algebraic expressions.
Multiplying Fractions
Multiplication of fractions, whether numerical or algebraic, follows a simple rule. The numerator of one fraction is multiplied with the numerator of the other, and similarly, the denominators are multiplied with each other. This results in a single fraction that represents the product of these two fractions.
Here鈥檚 a quick guide on multiplying fractions:
  • Multiply the numerators of the two fractions together to get the new numerator.
  • Multiply the denominators of the two fractions to get the new denominator.
  • Simplify the resulting fraction if possible, by looking for common factors.
In our example, multiplying the fractions \(\frac{(y-3)^2}{y-5} \) and \(\frac{y-5}{y-3} \) illustrates this principle well. The challenge lies not just in multiplying but also ensuring the resulting expression is in its simplest form.
Canceling Common Factors
Canceling common factors is like finding a shortcut in a complicated maze. It significantly simplifies the expressions by removing terms that appear in both the numerator and the denominator.
This process involves:
  • Identifying terms that occur in both the numerator and the denominator across the entire expression.
  • Eliminating these common terms from both parts of the fraction.
  • Simplifying what's left to arrive at the simplest form of the expression.
In our given problem, the factors \(y-5\) and \(y-3\) appear in both the numerator and the denominator of the multiplied expression. By canceling these terms carefully, we significantly reduce the complexity of the expression. This approach not only simplifies the computational effort but also helps in better understanding the relationships between different algebraic terms.

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