Problem 12
Question
For exercises 7-32, simplify. $$ \left(\frac{y^{2}+8 y}{y^{2}}\right)\left(\frac{9 y}{y+8}\right) $$
Step-by-Step Solution
Verified Answer
9
1Step 1: Simplify the first fraction
Simplify \ \ \ \ \(\frac{y^{2}+8 y}{y^{2}}\) \ \ \ \ by factoring out y in the numerator. The fraction becomes \(\frac{y(y + 8)}{y^2}\).
2Step 2: Cancel common factors in the first fraction
Cancel the common factor y in the numerator and denominator. The fraction simplifies to \(\frac{y + 8}{y}\).
3Step 3: Multiply the simplified fraction by the second fraction
Multiply \(\frac{y + 8}{y}\) by \(\frac{9 y}{y + 8}\). This becomes \(\left( \frac{y+8}{y} \right) \left( \frac{9y}{y+8} \right)\).
4Step 4: Cancel common factors in the product
Cancel the common factor \(y + 8\) in the numerator and denominator. The multiplication simplifies to \(\frac{9y}{y}\).
5Step 5: Simplify the final expression
Cancel the common factor y in the numerator and denominator, resulting in the final simplified expression: \(9\).
Key Concepts
FactoringCanceling Common FactorsNumerator and Denominator
Factoring
Factoring is a fundamental concept in algebra that involves breaking down an expression into simpler components, or factors, that when multiplied together give the original expression. In our exercise, we applied factoring to the first fraction \( \frac{y^{2}+8y}{y^{2}} \).
To do this, we looked at the numerator, \( y^{2}+8y \), and noticed that both terms share a common factor of \( y \). By factoring out \( y \), we rewrote the expression as \( y(y+8) \).
This process simplified the numerator, allowing us to make further simplifications later in solving the problem.
Remember: Always look for common terms you can factor out—this can greatly simplify your work and is often the key first step in solving algebraic expressions.
To do this, we looked at the numerator, \( y^{2}+8y \), and noticed that both terms share a common factor of \( y \). By factoring out \( y \), we rewrote the expression as \( y(y+8) \).
This process simplified the numerator, allowing us to make further simplifications later in solving the problem.
Remember: Always look for common terms you can factor out—this can greatly simplify your work and is often the key first step in solving algebraic expressions.
Canceling Common Factors
Canceling common factors involves simplifying fractions by removing terms that are present in both the numerator and the denominator. After factoring the numerator in our first fraction, we had \( \frac{y(y+8)}{y^{2}} \).
Notice that \( y \) is present in both the numerator and the denominator. By canceling out \( y \), we simplified the fraction to \( \frac{y+8}{y} \).
The same approach was used when multiplying the simplified fraction by the second fraction \( \frac{9y}{y+8} \). In the product \( \left( \frac{y+8}{y} \right) \left( \frac{9y}{y+8} \right) \), the term \( y+8 \) cancels out, reducing the expression to \( \frac{9y}{y} \).
Finally, we cancel the \( y \) terms to get the simplest form: \( 9 \).
Always check for common factors before multiplying or dividing fractions, as it can simplify your work and lead you to the correct answer faster.
Notice that \( y \) is present in both the numerator and the denominator. By canceling out \( y \), we simplified the fraction to \( \frac{y+8}{y} \).
The same approach was used when multiplying the simplified fraction by the second fraction \( \frac{9y}{y+8} \). In the product \( \left( \frac{y+8}{y} \right) \left( \frac{9y}{y+8} \right) \), the term \( y+8 \) cancels out, reducing the expression to \( \frac{9y}{y} \).
Finally, we cancel the \( y \) terms to get the simplest form: \( 9 \).
Always check for common factors before multiplying or dividing fractions, as it can simplify your work and lead you to the correct answer faster.
Numerator and Denominator
Understanding the roles of the numerator and denominator in fractions is crucial when simplifying algebraic expressions. The numerator is the top part of a fraction, while the denominator is the bottom part.
In the exercise, the first fraction was \( \frac{y^{2}+8y}{y^{2}} \). Here, \( y^{2}+8y \) is the numerator and \( y^{2} \) is the denominator. By factoring out common terms and canceling, we simplified the fraction.
When multiplying fractions, pay attention to how the numerators and denominators interact. For example, in \( \left( \frac{y+8}{y} \right) \left( \frac{9y}{y+8} \right) \), the \( y+8 \) terms in the numerator and denominator cancel each other out, simplifying the expression to \( \frac{9y}{y} \).
Minor adjustments to numerators and denominators, such as factoring and canceling, can lead to major simplifications in algebra.
In the exercise, the first fraction was \( \frac{y^{2}+8y}{y^{2}} \). Here, \( y^{2}+8y \) is the numerator and \( y^{2} \) is the denominator. By factoring out common terms and canceling, we simplified the fraction.
When multiplying fractions, pay attention to how the numerators and denominators interact. For example, in \( \left( \frac{y+8}{y} \right) \left( \frac{9y}{y+8} \right) \), the \( y+8 \) terms in the numerator and denominator cancel each other out, simplifying the expression to \( \frac{9y}{y} \).
Minor adjustments to numerators and denominators, such as factoring and canceling, can lead to major simplifications in algebra.
Other exercises in this chapter
Problem 12
For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{4 p}{9}}{\frac{2 p^{2}}{3}} $$
View solution Problem 12
For exercises \(5-48\), simplify. $$ \frac{6 w}{w+2}+\frac{w}{w+2} $$
View solution Problem 12
For exercises 1-66, simplify. $$ \frac{3 x-12}{15 x} $$
View solution Problem 13
The relationship of \(x\) and \(y\) is an inverse variation. When \(x=2, y=10\). a. Find the constant of proportionality, \(k\). b. Write an equation that repre
View solution