Find the highest power of each prime factor

Oct 13, 2025 by windailys.com 5 views

Introduction

When dealing with fractions, finding the lowest common denominator (LCD) is a crucial step in simplifying and adding or subtracting them. The LCD is the smallest multiple that both denominators can divide into evenly. In this article, we will explore how to find the LCD for the given set of fractions: $\frac{7}{x^2+4x+4} \cdot \frac{5}{4-x^2}$ .

Understanding the Denominators

To find the LCD, we first need to understand the denominators of the given fractions. The first denominator is $x^2+4x+4$ , which can be factored as $(x+2)^2$ . The second denominator is $4-x^2$ , which can be factored as $(2-x)(2+x)$ .

Factoring the Denominators

Let's factor the denominators to their prime factors:

$x^2+4x+4 = (x+2)^2$
$4-x^2 = (2-x)(2+x)$

Finding the LCD

To find the LCD, we need to find the product of the highest power of each prime factor that appears in either denominator. In this case, the prime factors are $(x+2)$ and $(2-x)$ and $(2+x)$ .

Calculating the LCD

The LCD is calculated by multiplying the highest power of each prime factor:

$(x+2)^2 = (x+2)(x+2)$
$(2-x)(2+x) = (2-x)(2+x)$

The LCD is the product of these two expressions:

$(x+2)(x+2)(2-x)(2+x)$

Simplifying the LCD

We can simplify the LCD by multiplying the expressions together:

$(x+2)(x+2)(2-x)(2+x) = (x^2+4x+4)(4-x^2)$

Expanding the expression, we get:

$(x^2+4x+4)(4-x^2) = 4x^2-4x^4+16x-16x^2+4x^2-4x^4$

Combining like terms, we get:

$-8x^4+12x^2+16x$

Conclusion

In conclusion, the lowest common denominator for the set of fractions $\frac{7}{x^2+4x+4} \cdot \frac{5}{4-x^2}$ is $-8x^4+12x^2+16x$ . This is the smallest multiple that both denominators can divide into evenly.

Frequently Asked Questions

What is the lowest common denominator? The lowest common denominator (LCD) is the smallest multiple that both denominators can divide into evenly.
How do I find the LCD? To find the LCD, you need to find the product of the highest power of each prime factor that appears in either denominator.
What is the formula for finding the LCD? The formula for finding the LCD is the product of the highest power of each prime factor.

Step-by-Step Guide to Finding the LCD

Factor the denominators: Factor the denominators to their prime factors.
Find the highest power of each prime factor: Find the highest power of each prime factor that appears in either denominator.
Multiply the highest power of each prime factor: Multiply the highest power of each prime factor to find the LCD.
Simplify the LCD: Simplify the LCD by combining like terms.

Common Mistakes to Avoid

Not factoring the denominators: Make sure to factor the denominators to their prime factors.
Not finding the highest power of each prime factor: Make sure to find the highest power of each prime factor that appears in either denominator.
Not multiplying the highest power of each prime factor: Make sure to multiply the highest power of each prime factor to find the LCD.

Real-World Applications

Finding the LCD is a crucial step in simplifying and adding or subtracting fractions. It is used in a variety of real-world applications, including:

Cooking: When measuring ingredients, fractions are often used. Finding the LCD is essential in simplifying and adding or subtracting fractions.
Finance: In finance, fractions are used to calculate interest rates and investment returns. Finding the LCD is essential in simplifying and adding or subtracting fractions.
Science: In science, fractions are used to calculate chemical reactions and measurements. Finding the LCD is essential in simplifying and adding or subtracting fractions.

Conclusion

In conclusion, finding the lowest common denominator is a crucial step in simplifying and adding or subtracting fractions. By following the steps outlined in this article, you can find the LCD for any set of fractions. Remember to factor the denominators, find the highest power of each prime factor, multiply the highest power of each prime factor, and simplify the LCD. With practice, you will become proficient in finding the LCD and be able to apply it to a variety of real-world applications.

Introduction

Finding the lowest common denominator (LCD) is a crucial step in simplifying and adding or subtracting fractions. However, it can be a challenging task, especially for those who are new to mathematics. In this article, we will answer some of the most frequently asked questions about finding the LCD.

Q&A

Q: What is the lowest common denominator?

A: The lowest common denominator (LCD) is the smallest multiple that both denominators can divide into evenly.

Q: How do I find the LCD?

A: To find the LCD, you need to find the product of the highest power of each prime factor that appears in either denominator.

Q: What is the formula for finding the LCD?

A: The formula for finding the LCD is the product of the highest power of each prime factor.

Q: Do I need to factor the denominators?

A: Yes, you need to factor the denominators to their prime factors before finding the LCD.

Q: How do I simplify the LCD?

A: To simplify the LCD, you need to combine like terms.

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 finding the highest power of each prime factor, and not multiplying the highest power of each prime factor.

Q: How do I apply the LCD in real-world applications?

A: The LCD is used in a variety of real-world applications, including cooking, finance, and science.

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

A: Yes, you can use a calculator to find the LCD. However, it is essential to understand the concept behind finding the LCD to use it effectively.

Q: How do I check if my answer is correct?

A: To check if your answer is correct, you need to verify that the LCD is the smallest multiple that both denominators can divide into evenly.

Q: Can I find the LCD for fractions with different signs?

A: Yes, you can find the LCD for fractions with different signs. However, you need to take into account the signs when multiplying the denominators.

Q: How do I find the LCD for fractions with variables?

A: To find the LCD for fractions with variables, you need to find the product of the highest power of each prime factor that appears in either denominator.

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

A: Yes, you can use the LCD to add or subtract fractions. However, you need to follow the rules of adding or subtracting fractions.

Conclusion

Finding the lowest common denominator is a crucial step in simplifying and adding or subtracting fractions. By understanding the concept behind finding the LCD and following the steps outlined in this article, you can find the LCD for any set of fractions. Remember to factor the denominators, find the highest power of each prime factor, multiply the highest power of each prime factor, and simplify the LCD. With practice, you will become proficient in finding the LCD and be able to apply it to a variety of real-world applications.

Additional Resources

Glossary

Lowest Common Denominator (LCD): The smallest multiple that both denominators can divide into evenly.
Prime Factor: A prime number that divides a number evenly.
Denominator: The bottom number of a fraction.
Numerator: The top number of a fraction.
Fraction: A way of expressing a part of a whole.