Problem 34
Question
For the following problems, perform the multiplications and divisions. $$ \frac{y+2}{2 y-1} \cdot \frac{2 y-1}{y-2} $$
Step-by-Step Solution
Verified Answer
Question: Multiply and simplify the given fractions: $\frac{(y+2)(2y-1)}{(2y-1)(y-2)}$.
Answer: The simplified result of multiplying and simplifying the given fractions is $\frac{y+2}{y-2}$.
1Step 1: Identify opportunities for cancellation
Before multiplication, check if there is any opportunity for cancellation. In this case, observe that the term \((2y-1)\) appears in both the numerator and the denominator. We can cancel this term out.
2Step 2: Perform the cancellation
After cancelling out the \((2y-1)\) terms, we are left with the following simplified expression:
$$
\frac{y+2}{1} \cdot \frac{1}{y-2}
$$
3Step 3: Multiply the fractions
Now, we will multiply the numerators together and the denominators together. In this case, the numerator will be \((y+2)\) and the denominator will be \((y-2)\).
$$
\frac{(y+2)}{(y-2)}
$$
4Step 4: Write down the final answer
The final answer after performing the multiplication and simplification of the given expression is:
$$
\frac{y+2}{y-2}
$$
Key Concepts
Fraction SimplificationCancellation LawFraction MultiplicationRational Expressions
Fraction Simplification
Simplifying fractions is a useful skill in algebra that helps make calculations easier. The goal is to reduce fractions to their simplest form by removing common factors from the numerator and the denominator.
When simplifying, it's important to identify any common factors or terms that appear in both the numerator and the denominator. For example, if you have a fraction like \( \frac{4}{8} \), you can simplify it to \( \frac{1}{2} \) by dividing both the top and bottom by the greatest common factor, which is 4 in this case.
This process ensures that the fraction is expressed in its simplest form, revealing its true value. Always aim to simplify fractions whenever possible to make your work more manageable, especially when dealing with larger algebraic expressions.
When simplifying, it's important to identify any common factors or terms that appear in both the numerator and the denominator. For example, if you have a fraction like \( \frac{4}{8} \), you can simplify it to \( \frac{1}{2} \) by dividing both the top and bottom by the greatest common factor, which is 4 in this case.
This process ensures that the fraction is expressed in its simplest form, revealing its true value. Always aim to simplify fractions whenever possible to make your work more manageable, especially when dealing with larger algebraic expressions.
Cancellation Law
Cancellation is a key concept in algebra, especially when working with fractions and rational expressions. It allows you to simplify complicated expressions by eliminating common factors from the numerator and the denominator.
The cancellation law states that if the same factor appears in both the numerator and denominator of a fraction, you can "cancel" it out. This is because any number divided by itself equals one.
For instance, in the expression \( \frac{2y-1}{2y-1} \), the \((2y-1)\) can be canceled out since \( (2y-1) / (2y-1) = 1 \). This simplification leads to a much neater expression and is fundamental when solving algebraic problems involving fractions. It is essential to always look for such opportunities to make calculations simpler.
The cancellation law states that if the same factor appears in both the numerator and denominator of a fraction, you can "cancel" it out. This is because any number divided by itself equals one.
For instance, in the expression \( \frac{2y-1}{2y-1} \), the \((2y-1)\) can be canceled out since \( (2y-1) / (2y-1) = 1 \). This simplification leads to a much neater expression and is fundamental when solving algebraic problems involving fractions. It is essential to always look for such opportunities to make calculations simpler.
Fraction Multiplication
Multiplying fractions involves a straightforward process. You multiply the numerators together and the denominators together. The result gives you a new fraction which may need further simplification.
For example, if you multiply \( \frac{a}{b} \) by \( \frac{c}{d} \), the result is \( \frac{a \cdot c}{b \cdot d} \). One thing to keep in mind is that it's often easier if you simplify the fractions before multiplying them. This can sometimes prevent the need for further simplification later on.
Also, remember to look for possible cancellation opportunities before and after multiplying. This can make your work easier and keeps the expressions as simple as possible. When fractions are in their simplest form, it is quicker to see relationships and solutions to the algebraic problems at hand.
For example, if you multiply \( \frac{a}{b} \) by \( \frac{c}{d} \), the result is \( \frac{a \cdot c}{b \cdot d} \). One thing to keep in mind is that it's often easier if you simplify the fractions before multiplying them. This can sometimes prevent the need for further simplification later on.
Also, remember to look for possible cancellation opportunities before and after multiplying. This can make your work easier and keeps the expressions as simple as possible. When fractions are in their simplest form, it is quicker to see relationships and solutions to the algebraic problems at hand.
Rational Expressions
Rational expressions are fractions where the numerator and the denominator are both polynomials. They follow similar rules to regular fractions but can be more complex due to the presence of variables and polynomial terms.
Working with rational expressions involves simplifying, multiplying, and dividing, just like with numerical fractions. Simplifying rational expressions usually involves factoring polynomials and looking for common factors or terms to cancel.
For instance, in the expression \( \frac{y^2 - 1}{y^2 - y - 2} \), you would factor both the numerator and the denominator and cancel out any common factors.
Understanding rational expressions is crucial for solving equations and inequalities in algebra. By mastering them, you unlock the ability to handle a wide range of problems involving multifaceted polynomial expressions.
Working with rational expressions involves simplifying, multiplying, and dividing, just like with numerical fractions. Simplifying rational expressions usually involves factoring polynomials and looking for common factors or terms to cancel.
For instance, in the expression \( \frac{y^2 - 1}{y^2 - y - 2} \), you would factor both the numerator and the denominator and cancel out any common factors.
Understanding rational expressions is crucial for solving equations and inequalities in algebra. By mastering them, you unlock the ability to handle a wide range of problems involving multifaceted polynomial expressions.
Other exercises in this chapter
Problem 34
For the following problems, solve the rational equations. $$ \frac{2 x-5}{x-6}=\frac{x+1}{x-6} $$
View solution Problem 34
For the following problems, show that the fractions are equivalent. $$ \frac{-2}{3} \text { and }-\frac{2}{3} $$
View solution Problem 34
For the following problems, add or subtract the rational expressions. $$ \frac{5 a+1}{a+7}+\frac{2 a-6}{a+7} $$
View solution Problem 34
For the following problems, reduce each rational expression to lowest terms. $$ \frac{(a+6)^{2}(a-7)^{6}}{(a+6)^{5}(a-7)^{2}} $$
View solution