Problem 33
Question
For the following problems, perform the multiplications and divisions. $$ \frac{2 x+5}{x+8} \cdot \frac{x+8}{x-2} $$
Step-by-Step Solution
Verified Answer
Question: Multiply and simplify the given rational expression: $$\frac{2x+5}{x+8} \cdot \frac{x+8}{x-2}.$$
Answer: The simplified expression is: $$\frac{2x^2 + 21x + 40}{x^2 + 6x - 16}.$$
1Step 1: Identify common factors
In the given expression, $$\frac{2x+5}{x+8} \cdot \frac{x+8}{x-2},$$ we can notice that \((x+8)\) is a common factor between the numerator and denominator.
2Step 2: Multiply numerators
Now, we multiply the numerators of the two fractions:
$$(2x+5) \cdot (x+8) = 2x^2 + 16x + 5x + 40 = 2x^2 + 21x + 40.$$
3Step 3: Multiply denominators
Next, multiply the denominators of the two fractions:
$$(x+8) \cdot (x-2) = x^2 - 2x + 8x - 16 = x^2 + 6x - 16.$$
4Step 4: Combine numerators and denominators
Now, we combine the products of the numerators and denominators into a single fraction:
$$\frac{2x^2 + 21x + 40}{x^2 + 6x - 16}.$$
5Step 5: Cancel common factors (if any)
In this case, we do not have any other common factors between the numerator and denominator, so the simplified expression is:
$$\frac{2x^2 + 21x + 40}{x^2 + 6x - 16}.$$
Key Concepts
Multiplication of FractionsDivision of FractionsSimplifying Fractions
Multiplication of Fractions
Multiplying fractions might seem tricky, but it’s quite simple when broken down. The key idea is to multiply the numerators together and the denominators together.
Think of fractions as numbers over numbers. To multiply \(\frac{a}{b}\) by \(\frac{c}{d}\), you simply do the following:
This leads to the new fraction \(\frac{a \times c}{b \times d}\). The same logic applies to our algebraic expression, \(\frac{2x+5}{x+8} \cdot \frac{x+8}{x-2}\).
First, notice the common factor \((x+8)\), but the method remains consistent. Multiply the numerators \((2x+5) \cdot (x+8)\) and the denominators \((x+8) \cdot (x-2)\).
This results in: \[\frac{2x^2+21x+40}{x^2+6x-16}\].
Multiplication is all about combining like parts, and the process helps set up further simplification.
Think of fractions as numbers over numbers. To multiply \(\frac{a}{b}\) by \(\frac{c}{d}\), you simply do the following:
- Multiply the numerators: \(a \times c\)
- Multiply the denominators: \(b \times d\)
This leads to the new fraction \(\frac{a \times c}{b \times d}\). The same logic applies to our algebraic expression, \(\frac{2x+5}{x+8} \cdot \frac{x+8}{x-2}\).
First, notice the common factor \((x+8)\), but the method remains consistent. Multiply the numerators \((2x+5) \cdot (x+8)\) and the denominators \((x+8) \cdot (x-2)\).
This results in: \[\frac{2x^2+21x+40}{x^2+6x-16}\].
Multiplication is all about combining like parts, and the process helps set up further simplification.
Division of Fractions
Dividing fractions involves a small twist compared to multiplication. Instead of directly dividing, you multiply by the reciprocal.
Let’s break it down: When dividing \(\frac{a}{b}\) by \(\frac{c}{d}\), you flip the second fraction (the divisor) to get \(\frac{d}{c}\), then multiply:
This results in \(\frac{a \times d}{b \times c}\). In contexts like the given problem, fractions can switch roles via multiplication and division interplays, but understanding this concept firmly is crucial for dealing with more intricate expressions, blending multiplication with division.
Let’s break it down: When dividing \(\frac{a}{b}\) by \(\frac{c}{d}\), you flip the second fraction (the divisor) to get \(\frac{d}{c}\), then multiply:
- Flip \(\frac{c}{d}\) to \(\frac{d}{c}\)
- Multiply \(\frac{a}{b} \times \frac{d}{c}\)
This results in \(\frac{a \times d}{b \times c}\). In contexts like the given problem, fractions can switch roles via multiplication and division interplays, but understanding this concept firmly is crucial for dealing with more intricate expressions, blending multiplication with division.
Simplifying Fractions
Simplifying fractions is like cleaning them up. It makes fractions easier to understand by reducing them to their smallest form.
When it comes to algebraic fractions, simplification can involve canceling common factors in the numerator and the denominator. Look out for terms that appear in both.
For example, if you find a common term such as \((x+8)\) in both the numerator and denominator like we did, they cancel out, simplifying the expression.
After performing operations like multiplication or division, the expression becomes a single fraction. The next step is always checking if there are common factors. If there aren’t any, that’s your answer.
Simplifying makes it easier to work with fractions in any future calculations. It also reveals the true ‘face’ of the fraction: its simplest form.
Follow these steps to ensure clear and correct results in algebra.
When it comes to algebraic fractions, simplification can involve canceling common factors in the numerator and the denominator. Look out for terms that appear in both.
For example, if you find a common term such as \((x+8)\) in both the numerator and denominator like we did, they cancel out, simplifying the expression.
After performing operations like multiplication or division, the expression becomes a single fraction. The next step is always checking if there are common factors. If there aren’t any, that’s your answer.
Simplifying makes it easier to work with fractions in any future calculations. It also reveals the true ‘face’ of the fraction: its simplest form.
Follow these steps to ensure clear and correct results in algebra.
Other exercises in this chapter
Problem 33
For the following problems, solve the rational equations. $$ \frac{3 a-7}{a-3}=\frac{4 a-10}{a-3} $$
View solution Problem 33
For the following problems, show that the fractions are equivalent. $$ -\frac{1}{4} \text { and } \frac{-1}{4} $$
View solution Problem 33
For the following problems, replace \(N\) with the proper quantity. $$ \frac{2}{a^{2}}=\frac{N}{a^{2}(a-1)} $$
View solution Problem 33
For the following problems, add or subtract the rational expressions. $$ \frac{a+6}{a-1}+\frac{3 a+5}{a-1} $$
View solution