Find The Least Common Denominator Of $\frac{x}{3x-6}$ And $\frac{2}{7x-14}$.
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Introduction
In mathematics, the least common denominator (LCD) is an essential concept in algebra, particularly when dealing with rational expressions. It is the smallest multiple that two or more denominators have in common. In this article, we will explore how to find the least common denominator of two rational expressions: $\frac{x}{3x-6}$ and $\frac{2}{7x-14}$.
Understanding Rational Expressions
A rational expression is a fraction that contains variables and/or constants in the numerator and/or denominator. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular fractions. However, when dealing with rational expressions, we must be careful to ensure that the denominators are not equal to zero.
Simplifying the Rational Expressions
Before finding the least common denominator, let's simplify the given rational expressions.
Simplifying $\frac{x}{3x-6}$
We can simplify the expression $\frac{x}{3x-6}$ by factoring the denominator.
Simplifying $\frac{2}{7x-14}$
We can simplify the expression $\frac{2}{7x-14}$ by factoring the denominator.
Finding the Least Common Denominator
Now that we have simplified the rational expressions, we can find the least common denominator.
The least common denominator is the smallest multiple that the denominators have in common. In this case, the denominators are $3(x-2)$ and $7(x-2)$.
To find the least common denominator, we need to find the smallest multiple that $3(x-2)$ and $7(x-2)$ have in common.
Finding the Least Common Multiple
The least common multiple (LCM) of two numbers is the smallest number that both numbers divide into evenly.
In this case, the LCM of $3(x-2)$ and $7(x-2)$ is $21(x-2)$.
The Least Common Denominator
The least common denominator is the LCM of the denominators. In this case, the least common denominator is $21(x-2)$.
Conclusion
In conclusion, the least common denominator of $\frac{x}{3x-6}$ and $\frac{2}{7x-14}$ is $21(x-2)$. This is the smallest multiple that the denominators have in common.
Example Use Case
The least common denominator is an essential concept in algebra, particularly when dealing with rational expressions. It is used to add, subtract, multiply, and divide rational expressions.
For example, suppose we want to add the rational expressions $\frac{x}{3x-6}$ and $\frac{2}{7x-14}$.
To add the rational expressions, we need to find the least common denominator, which is $21(x-2)$.
We can then rewrite the rational expressions with the least common denominator.
We can then add the rational expressions.
Final Answer
The final answer is:
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Introduction
In our previous article, we discussed how to find the least common denominator (LCD) of two rational expressions: $\frac{x}{3x-6}$ and $\frac{2}{7x-14}$. In this article, we will answer some frequently asked questions about finding the LCD of rational expressions.
Q&A
Q: What is the least common denominator?
A: The least common denominator (LCD) is the smallest multiple that two or more denominators have in common.
Q: How do I find the least common denominator?
A: To find the LCD, you need to find the smallest multiple that the denominators have in common. You can do this by factoring the denominators and finding the least common multiple (LCM) of the factors.
Q: What is the difference between the least common denominator and the least common multiple?
A: The least common denominator (LCD) is the smallest multiple that the denominators have in common, while the least common multiple (LCM) is the smallest number that both numbers divide into evenly.
Q: Can I find the least common denominator of more than two rational expressions?
A: Yes, you can find the LCD of more than two rational expressions. To do this, you need to find the LCD of the first two expressions, and then find the LCD of the result and the third expression.
Q: How do I use the least common denominator to add, subtract, multiply, and divide rational expressions?
A: To add, subtract, multiply, and divide rational expressions, you need to find the LCD of the expressions and then rewrite the expressions with the LCD.
Q: What if the denominators are not factorable?
A: If the denominators are not factorable, you can use the prime factorization method to find the LCD.
Q: Can I use a calculator to find the least common denominator?
A: Yes, you can use a calculator to find the LCD. However, it's always a good idea to understand the concept and be able to do it by hand.
Example Use Cases
Example 1: Finding the LCD of Two Rational Expressions
Suppose we want to find the LCD of $\frac{x}{3x-6}$ and $\frac{2}{7x-14}$.
To find the LCD, we need to find the smallest multiple that the denominators have in common. We can do this by factoring the denominators and finding the LCM of the factors.
The LCM of $3(x-2)$ and $7(x-2)$ is $21(x-2)$.
Therefore, the LCD of $\frac{x}{3x-6}$ and $\frac{2}{7x-14}$ is $21(x-2)$.
Example 2: Using the LCD to Add Rational Expressions
Suppose we want to add the rational expressions $\frac{x}{3x-6}$ and $\frac{2}{7x-14}$.
To add the rational expressions, we need to find the LCD, which is $21(x-2)$.
We can then rewrite the rational expressions with the LCD.
We can then add the rational expressions.
Final Answer
The final answer is: