Problem 38
Question
Multiply. $$\frac{8 x-12}{14 x+7} \cdot \frac{42 x+21}{32 x-48}$$
Step-by-Step Solution
Verified Answer
\(\frac{3}{4}\)
1Step 1: Simplify Fractions
The first manipulation is to simplify the fractions if possible. \( \frac{8x-12}{14x+7} \) simplifies to \( \frac{4(2x-3)}{7(2x+1)} \) and \( \frac{42x+21}{32x-48} \) simplifies to \( \frac{21(2x+1)}{16(2x-3)} \)
2Step 2: Multiply Numerators and Denominators
Afterwards, multiply the two simplified fractions together like one would do for any other fractions. Here, the numerators of the two fractions are multiplied together to form the new numerator and the denominators are also multiplied to form the new denominator. \( \frac{4(2x-3)}{7(2x+1)} \cdot \frac{21(2x+1)}{16(2x-3)} = \frac{4*21*(2x-3)*(2x+1)}{7*16*(2x+1)*(2x-3)} \)
3Step 3: Simplify Resulting Fraction
Lastly, simplify the resulting fraction by canceling out the same elements in the numerator and denominator. \( \frac{4*21*(2x-3)*(2x+1)}{7*16*(2x+1)*(2x-3)} \) simplifies to \( \frac{21}{28} \) or \( \frac{3}{4} \)
Key Concepts
Simplifying FractionsMultiplying FractionsFactorizationFraction Simplification Steps
Simplifying Fractions
Simplifying fractions is all about making them look simpler without changing their value. To simplify, look for a number or expression that can divide both the numerator and the denominator evenly. In algebraic fractions, this often involves spotting factors that are common in both. For instance, in the exercise, one fraction simplifies from \( \frac{8x-12}{14x+7} \) to \( \frac{4(2x-3)}{7(2x+1)} \). This is done by factoring the numerator and the denominator.
Factoring means we have to express numbers or expressions as a product of their factors. Here are steps to simplify algebraic fractions:
Factoring means we have to express numbers or expressions as a product of their factors. Here are steps to simplify algebraic fractions:
- Factor out the greatest common factor (GCF) from the numerator and the denominator.
- After factoring both parts, cancel out any identical factors that appear in both. This reduces the fraction effectively.
Multiplying Fractions
To multiply fractions, multiply the numerators across to get the new numerator and the denominators across to get the new denominator. In our example, we found the simplified form of the product by multiplying:
However, always look for simplification steps afterward. Many times, multiplying results in higher numbers or complex expressions that can be simplified, ensuring the answer is as compact as possible.
- Numerators: \( 4(2x-3) \times 21(2x+1) \)
- Denominators: \( 7(2x+1) \times 16(2x-3) \)
However, always look for simplification steps afterward. Many times, multiplying results in higher numbers or complex expressions that can be simplified, ensuring the answer is as compact as possible.
Factorization
Factorization is a key component of simplifying algebraic fractions. It involves breaking down an expression into products of simpler expressions or numbers. For the given equation, this is crucial because simplifying relies on canceling common factors.
Here's how it works in our example: The expression \( 8x-12 \) factors to \( 4(2x-3) \), and \( 14x+7 \) factors to \( 7(2x+1) \). Recognizing these patterns helps not only to simplify the fraction but also in multiplying the expression successfully. Factorizing each term separately is essential. Once broken down, equivalent terms in the numerator and denominator can be canceled, leading to a simpler fraction.
Practice recognizing factorable patterns such as differences of squares, quadratic trinomials, and common monomial factors. This skill reduces complexity quickly and accurately.
Here's how it works in our example: The expression \( 8x-12 \) factors to \( 4(2x-3) \), and \( 14x+7 \) factors to \( 7(2x+1) \). Recognizing these patterns helps not only to simplify the fraction but also in multiplying the expression successfully. Factorizing each term separately is essential. Once broken down, equivalent terms in the numerator and denominator can be canceled, leading to a simpler fraction.
Practice recognizing factorable patterns such as differences of squares, quadratic trinomials, and common monomial factors. This skill reduces complexity quickly and accurately.
Fraction Simplification Steps
Simplifying fractions involves recognizing and following a sequence of logical steps. Here’s a brief guide you can follow:
In our initial example, after multiplying the simplified fractions, further simplification led to reducing \( \frac{21}{28} \) to \( \frac{3}{4} \), as one common factor of 7 existed in both. These steps not only ensure clarity in mathematical expressions but also build a strong foundation for dealing with more complex algebraic expressions.
Applying these steps consistently builds accuracy and confidence while working with algebraic fractions.
- Identify and separate the greatest common factors from both the numerator and the denominator.
- Express the fraction in its factored form.
- Cancel identical factors in both the numerator and the denominator.
- Simplify the resulting fraction as much as possible.
In our initial example, after multiplying the simplified fractions, further simplification led to reducing \( \frac{21}{28} \) to \( \frac{3}{4} \), as one common factor of 7 existed in both. These steps not only ensure clarity in mathematical expressions but also build a strong foundation for dealing with more complex algebraic expressions.
Applying these steps consistently builds accuracy and confidence while working with algebraic fractions.
Other exercises in this chapter
Problem 38
Solve. $$\frac{x}{x+4}=\frac{11}{x^{2}-16}+2$$
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Simplify. $$1+\frac{1}{1+\frac{1}{2}}$$
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A camper drove 80 mi to a recreational area and then hiked 4 mi into the woods. The rate of the camper while driving was ten times the rate while hiking. The to
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