Problem 77
Question
For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{1}{x-7}, \frac{4}{x-1} $$
Step-by-Step Solution
Verified Answer
Answer: The converted rational expressions with the same denominators are:
$$
\frac{x-1}{(x-7)(x-1)} \quad \text{and} \quad \frac{4(x-7)}{(x-1)(x-7)}
$$
1Step 1: Identify the given rational expressions
The given rational expressions are:
$$
\frac{1}{x-7} \quad \text{and} \quad \frac{4}{x-1}
$$
2Step 2: Find the least common denominator (LCD)
The least common denominator is the smallest expression that can be evenly divided by both denominators. In this case, since the denominators have no common factors other than 1, the LCD is simply the product of the two denominators:
$$
\text{LCD} = (x-7)(x-1)
$$
3Step 3: Rewrite each fraction with the LCD as the new denominator
In order to do this, we will multiply each fraction by a fraction that has a value of 1, but adjusts the denominator to be the LCD.
For the first fraction, we will multiply it by \(\displaystyle \frac{x-1}{x-1}\):
$$
\frac{1}{x-7} \times \frac{x-1}{x-1} = \frac{(x-1)}{(x-7)(x-1)}
$$
For the second fraction, we will multiply it by \(\displaystyle \frac{x-7}{x-7}\):
$$
\frac{4}{x-1} \times \frac{x-7}{x-7} = \frac{4(x-7)}{(x-1)(x-7)}
$$
4Step 4: Present the converted rational expressions
The rational expressions converted to have the same denominators are:
$$
\frac{x-1}{(x-7)(x-1)} \quad \text{and} \quad \frac{4(x-7)}{(x-1)(x-7)}
$$
Key Concepts
Least Common DenominatorAlgebraic FractionsConverting Denominators
Least Common Denominator
Understanding the concept of the least common denominator (LCD) is essential when working with rational expressions, particularly when you need to add, subtract, or compare fractions. The LCD is the smallest expression that both denominators of given fractions can divide into without leaving a remainder. Imagine it as the 'meeting ground' for different fractions to ensure they can be easily combined.
When the denominators are different and there are no common factors, the LCD is simply the product of these two denominators. Consider fractions with denominators that are polynomials. To identify the LCD, you should first factor each denominator, if possible, to find common terms. However, in our given exercise, the denominators are \(x-7\) and \(x-1\), which are already in their simplest form and share no common factors. Therefore, the LCD is their product \( (x-7)(x-1) \).
This concept serves as a foundation for many algebraic operations involving fractions, and a proper grasp of the LCD can greatly simplify complex expressions.
When the denominators are different and there are no common factors, the LCD is simply the product of these two denominators. Consider fractions with denominators that are polynomials. To identify the LCD, you should first factor each denominator, if possible, to find common terms. However, in our given exercise, the denominators are \(x-7\) and \(x-1\), which are already in their simplest form and share no common factors. Therefore, the LCD is their product \( (x-7)(x-1) \).
This concept serves as a foundation for many algebraic operations involving fractions, and a proper grasp of the LCD can greatly simplify complex expressions.
Algebraic Fractions
Algebraic fractions are simply fractions where the numerator, denominator, or both contain algebraic expressions. Similar to numerical fractions, they adhere to the same arithmetic rules, but the presence of variables introduces additional considerations for simplification and computation.
It's crucial when dealing with algebraic fractions, as in our exercise \(\frac{1}{x-7}\) and \(\frac{4}{x-1}\), to maintain the integrity of the expressions by avoiding any alteration of their values. This involves techniques such as expanding, factoring, and reducing. In the solution's Step 3, converting the fractions to have the same denominator requires multiplying by an intelligent form of 1 (a fraction with equal numerator and denominator that don't change the fraction's value), preserving the value while achieving a common denominator.
It's crucial when dealing with algebraic fractions, as in our exercise \(\frac{1}{x-7}\) and \(\frac{4}{x-1}\), to maintain the integrity of the expressions by avoiding any alteration of their values. This involves techniques such as expanding, factoring, and reducing. In the solution's Step 3, converting the fractions to have the same denominator requires multiplying by an intelligent form of 1 (a fraction with equal numerator and denominator that don't change the fraction's value), preserving the value while achieving a common denominator.
Converting Denominators
Converting denominators to a common value is a crucial process when managing rational expressions, particularly for operations like addition or comparison. This process involves making the denominators the same without changing the value of the fractions involved.
Take our exercise for instance. To convert \(\frac{1}{x-7}\) and \(\frac{4}{x-1}\) to have the same denominator, we looked at the denominators and multiplied each fraction by a form of 1 that adjusted its denominator to the LCD. For the first fraction, multiplying by \(\frac{x-1}{x-1}\) and for the second by \(\frac{x-7}{x-7}\) served precisely this purpose. Post multiplication, we obtain expressions with matching denominators, allowing for straightforward subsequent mathematical operations.
This conversion process is a powerful tool that enables us to manipulate and simplify algebraic fractions, a necessity for any student looking to master algebra.
Take our exercise for instance. To convert \(\frac{1}{x-7}\) and \(\frac{4}{x-1}\) to have the same denominator, we looked at the denominators and multiplied each fraction by a form of 1 that adjusted its denominator to the LCD. For the first fraction, multiplying by \(\frac{x-1}{x-1}\) and for the second by \(\frac{x-7}{x-7}\) served precisely this purpose. Post multiplication, we obtain expressions with matching denominators, allowing for straightforward subsequent mathematical operations.
This conversion process is a powerful tool that enables us to manipulate and simplify algebraic fractions, a necessity for any student looking to master algebra.
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Problem 77
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