Problem 48
Question
Simplify the rational expression. $$ \frac{\frac{2 x}{3}+\frac{4 x}{7}}{\frac{x}{2}} $$
Step-by-Step Solution
Verified Answer
The simplified expression is \(\frac{52}{21}\).
1Step 1: Simplify the Numerator
To simplify the numerator of \(\frac{2x}{3} + \frac{4x}{7}\), we need to find a common denominator. The denominators are 3 and 7, so the common denominator is 21. Convert each fraction: \(\frac{2x}{3} = \frac{14x}{21}\) and \(\frac{4x}{7} = \frac{12x}{21}\). Add them: \(\frac{14x}{21} + \frac{12x}{21} = \frac{26x}{21}\). Thus, the simplified numerator is \(\frac{26x}{21}\).
2Step 2: Write the Full Expression
Now that the numerator is simplified, rewrite the entire expression: \(\frac{\frac{26x}{21}}{\frac{x}{2}}\).
3Step 3: Simplify the Division of Fractions
To simplify the division of two fractions, multiply the numerator by the reciprocal of the denominator: \(\frac{26x}{21} \times \frac{2}{x}\).
4Step 4: Simplify the Expression
Multiply the fractions: \(\frac{26x \cdot 2}{21 \cdot x} = \frac{52x}{21x}\). Cancel the \(x\) terms in the numerator and denominator: \(\frac{52}{21}\).
5Step 5: Simplify the Fraction Further if Possible
Check and simplify \(\frac{52}{21}\) as much as possible. Since 52 and 21 have no common factors besides 1, the expression is already in its simplest form.
Key Concepts
SimplificationFractionsCommon Denominator
Simplification
Simplification is all about making mathematical expressions easier to understand and work with. Take the initial expression as an example: \( \frac{\frac{2x}{3} + \frac{4x}{7}}{\frac{x}{2}} \).
The goal is to simplify this expression to a more manageable form.
When you simplify an expression, you aim to reduce it to its most basic form without changing its value. This involves combining like terms, using common denominators for fractions, and canceling terms when possible.
In this exercise, we first simplified the numerator by finding a common denominator, which reduces confusion and allows for straightforward arithmetic operations.
Further, we simplified a complex fraction by turning it into a multiplication problem. This is often easier than dealing with division directly.
Remember, simplification is not about changing what an expression represents; it's about finding the clearest and most efficient way to present it.
The goal is to simplify this expression to a more manageable form.
When you simplify an expression, you aim to reduce it to its most basic form without changing its value. This involves combining like terms, using common denominators for fractions, and canceling terms when possible.
In this exercise, we first simplified the numerator by finding a common denominator, which reduces confusion and allows for straightforward arithmetic operations.
Further, we simplified a complex fraction by turning it into a multiplication problem. This is often easier than dealing with division directly.
Remember, simplification is not about changing what an expression represents; it's about finding the clearest and most efficient way to present it.
Fractions
Fractions are a way of representing numbers that are not whole. They are made up of two parts:
Adding fractions requires them to have a common denominator. This involves finding a common multiple of the denominators and adjusting the numerators accordingly.
For example, the denominators 3 and 7 both multiply to 21, allowing \( \frac{2x}{3} \) to become \( \frac{14x}{21} \) and \( \frac{4x}{7} \) to become \( \frac{12x}{21} \). This adjustment makes it straightforward to add them directly.
Proper understanding of fractions and how they interact when added, subtracted, or divided is foundational to working with rational expressions.
- Numerator: The top part of the fraction that represents how many parts are being considered.
- Denominator: The bottom part of the fraction that indicates how many parts make up a whole.
Adding fractions requires them to have a common denominator. This involves finding a common multiple of the denominators and adjusting the numerators accordingly.
For example, the denominators 3 and 7 both multiply to 21, allowing \( \frac{2x}{3} \) to become \( \frac{14x}{21} \) and \( \frac{4x}{7} \) to become \( \frac{12x}{21} \). This adjustment makes it straightforward to add them directly.
Proper understanding of fractions and how they interact when added, subtracted, or divided is foundational to working with rational expressions.
Common Denominator
Finding a common denominator is a critical step in simplifying rational expressions.
It allows us to combine fractions with different denominators by converting them to equivalent fractions with the same denominator.
In the exercise, the fractions \( \frac{2x}{3} \) and \( \frac{4x}{7} \) needed a shared basis for addition. 21 was chosen as it is a common multiple of both 3 and 7.
This conversion \( \frac{2x}{3} \to \frac{14x}{21} \), \( \frac{4x}{7} \to \frac{12x}{21} \) requires multiplying both the numerator and denominator of each fraction by the necessary amount to achieve 21 in the denominator.
Once this is achieved, the fractions become easy to add: \( \frac{14x}{21} + \frac{12x}{21} = \frac{26x}{21} \).
This technique of aligning denominators by using the smallest common multiple simplifies calculations and is a powerful tool in managing both simple and complex calculations involving fractions.
It allows us to combine fractions with different denominators by converting them to equivalent fractions with the same denominator.
In the exercise, the fractions \( \frac{2x}{3} \) and \( \frac{4x}{7} \) needed a shared basis for addition. 21 was chosen as it is a common multiple of both 3 and 7.
This conversion \( \frac{2x}{3} \to \frac{14x}{21} \), \( \frac{4x}{7} \to \frac{12x}{21} \) requires multiplying both the numerator and denominator of each fraction by the necessary amount to achieve 21 in the denominator.
Once this is achieved, the fractions become easy to add: \( \frac{14x}{21} + \frac{12x}{21} = \frac{26x}{21} \).
This technique of aligning denominators by using the smallest common multiple simplifies calculations and is a powerful tool in managing both simple and complex calculations involving fractions.
Other exercises in this chapter
Problem 48
For the following exercises, find the sum or difference. $$(9 m+4 n-1)(2 m+8)$$
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For the following exercises, multiply the polynomials. $$ (9 m+4 n-1)(2 m+8) $$
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Simplify each expression. $$q^{\frac{3}{2}} \sqrt{63 p}$$
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The Gross Domestic Product (GDP) for the United States in the first quarter of 2014 was \(\$ 1.71496 \times 10^{13}\) . Rewrite the GDP in standard notation.
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