Find The LCD Of The Following. Do Not Combine Fractions.\[$\frac{3}{5x - 10} \cdot \frac{5}{6x - 12}\$\]The LCD Is \[$\square\$\].

Understanding the Concept of LCD

In mathematics, the least common denominator (LCD) is the smallest multiple that two or more denominators have in common. It is an essential concept in algebra, particularly when working with rational expressions. In this article, we will explore how to find the LCD of a given rational expression.

The Given Rational Expression

The given rational expression is:

$\frac{3}{5x - 10} \cdot \frac{5}{6x - 12}$

Step 1: Factor the Denominators

To find the LCD, we need to factor the denominators of the given rational expression. Let's start by factoring the first denominator, $5x - 10$.

$5x - 10 = 5(x - 2)$

Now, let's factor the second denominator, $6x - 12$.

$6x - 12 = 6(x - 2)$

Step 2: Identify the Common Factors

Now that we have factored the denominators, we can identify the common factors. In this case, both denominators have a common factor of $(x - 2)$.

Step 3: Find the LCD

To find the LCD, we need to multiply the common factors together. In this case, the LCD is:

$5(x - 2) \cdot 6(x - 2) = 30(x - 2)^2$

The Final Answer

Therefore, the least common denominator (LCD) of the given rational expression is:

$\boxed{30(x - 2)^2}$

Why is the LCD Important?

The LCD is an essential concept in algebra because it allows us to simplify rational expressions and perform operations such as addition and subtraction. By finding the LCD, we can rewrite the rational expression with a common denominator, making it easier to work with.

Real-World Applications

The concept of LCD has real-world applications in various fields, including science, engineering, and economics. For example, in physics, the LCD is used to calculate the frequency of a wave. In engineering, the LCD is used to design electronic circuits. In economics, the LCD is used to calculate the interest rate of a loan.

Conclusion

In conclusion, finding the least common denominator (LCD) of a rational expression is an essential concept in algebra. By following the steps outlined in this article, we can find the LCD of a given rational expression. The LCD is an important concept that has real-world applications in various fields.

Common Mistakes to Avoid

When finding the LCD, there are several common mistakes to avoid. These include:

• Not factoring the denominators
• Not identifying the common factors
• Not multiplying the common factors together
• Not checking for any other common factors

Tips and Tricks

Here are some tips and tricks to help you find the LCD:

• Always factor the denominators
• Identify the common factors by looking for any common factors between the two denominators
• Multiply the common factors together
• Check for any other common factors
• Use a calculator to check your answer

Practice Problems

Here are some practice problems to help you practice finding the LCD:

• Find the LCD of $\frac{2}{x + 1} \cdot \frac{3}{x + 2}$
• Find the LCD of $\frac{4}{x - 2} \cdot \frac{5}{x - 3}$
• Find the LCD of $\frac{6}{x + 4} \cdot \frac{7}{x + 5}$

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about finding the least common denominator (LCD) of a rational expression.

Q: What is the least common denominator (LCD)?

A: The least common denominator (LCD) is the smallest multiple that two or more denominators have in common. It is an essential concept in algebra, particularly when working with rational expressions.

Q: Why is the LCD important?

A: The LCD is an essential concept in algebra because it allows us to simplify rational expressions and perform operations such as addition and subtraction. By finding the LCD, we can rewrite the rational expression with a common denominator, making it easier to work with.

Q: How do I find the LCD of a rational expression?

A: To find the LCD of a rational expression, you need to follow these steps:

1. Factor the denominators
2. Identify the common factors
3. Multiply the common factors together

Q: What if the denominators do not have any common factors?

A: If the denominators do not have any common factors, then the LCD is the product of the two denominators.

Q: Can I use a calculator to find the LCD?

A: Yes, you can use a calculator to find the LCD. However, it is always a good idea to check your answer by hand to make sure it is correct.

Q: What are some common mistakes to avoid when finding the LCD?

A: Some common mistakes to avoid when finding the LCD include:

• Not factoring the denominators
• Not identifying the common factors
• Not multiplying the common factors together
• Not checking for any other common factors

Q: How do I check my answer for the LCD?

A: To check your answer for the LCD, you can multiply the numerator and denominator of the rational expression by the LCD and simplify.

Q: Can I use the LCD to add or subtract rational expressions?

A: Yes, you can use the LCD to add or subtract rational expressions. By rewriting the rational expressions with a common denominator, you can add or subtract the numerators and keep the denominator the same.

Q: What are some real-world applications of the LCD?

A: The LCD has real-world applications in various fields, including science, engineering, and economics. For example, in physics, the LCD is used to calculate the frequency of a wave. In engineering, the LCD is used to design electronic circuits. In economics, the LCD is used to calculate the interest rate of a loan.

Q: Can I find the LCD of a rational expression with multiple terms?

A: Yes, you can find the LCD of a rational expression with multiple terms. To do this, you need to factor each term and identify the common factors.

Q: What if the rational expression has a variable in the denominator?

A: If the rational expression has a variable in the denominator, you need to factor the denominator and identify the common factors.

Conclusion

In conclusion, finding the least common denominator (LCD) of a rational expression is an essential concept in algebra. By following the steps outlined in this article, we can find the LCD of a given rational expression. The LCD is an important concept that has real-world applications in various fields.

Practice Problems

Here are some practice problems to help you practice finding the LCD:

• Find the LCD of $\frac{2}{x + 1} \cdot \frac{3}{x + 2}$
• Find the LCD of $\frac{4}{x - 2} \cdot \frac{5}{x - 3}$
• Find the LCD of $\frac{6}{x + 4} \cdot \frac{7}{x + 5}$

Additional Resources

For more information on finding the LCD, you can check out the following resources:

• Khan Academy: Least Common Denominator (LCD)
• Mathway: Least Common Denominator (LCD)
• Wolfram Alpha: Least Common Denominator (LCD)

Conclusion

In conclusion, finding the least common denominator (LCD) of a rational expression is an essential concept in algebra. By following the steps outlined in this article, we can find the LCD of a given rational expression. The LCD is an important concept that has real-world applications in various fields.