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Prime factorization is a method used to express a number as the product of its prime factors. Primes are numbers that only have two positive divisors: 1 and themselves. For example, to factorize 15, you break it down into 3 and 5鈥攂oth prime numbers, so 15 becomes 3 脳 5. Similarly, 36 can be factored as 2 脳 2 脳 3 脳 3 or in exponential form as 2虏 脳 3虏. This step is crucial because it allows us to see the underlying structure of the numbers. For our exercise, the factorization lets us identify all the prime factors involved in the denominators, which is essential for finding the Least Common Denominator (LCD). Understanding the individual components makes it simpler to combine the highest powers of each element in later steps.

Once we have the prime factorizations, the next step is to compare the denominators. In our example, the denominators are 15y虏 and 36y鈦�. First, focus on the numerical coefficients: 15 is 3 脳 5, and 36 is 2虏 脳 3虏. Next, incorporate the variable parts: y虏 and y鈦�. By isolating and understanding each component鈥攂oth the numerical coefficients and the variable parts鈥攚e can more easily see what needs to be combined to find the LCD. This comparison highlights the importance of being meticulous in identifying each element that constitutes the denominators.

In order to find the Least Common Denominator (LCD), we need to combine the highest powers of each prime factor present in the denominators. Let鈥檚 break it down: For the prime number 2, the highest power we need is 2虏 (from 36). For the prime number 3, the highest power is 3虏 (also from 36). For the prime number 5, the highest power is 5 (from 15). For the variable y, the highest power present is y鈦�. By collecting these highest powers, we ensure that both denominators can be evenly divided into the LCD. Combining them correctly is the next critical step.

To find the LCD, we multiply the highest powers of all the prime factors and variable parts identified. In our example, we take 2虏, 3虏, 5, and y鈦� and multiply them together:

• 2虏 = 4
• 3虏 = 9
• 5 = 5
• y鈦� = y鈦�

Thus, the product is: 4 脳 9 脳 5 脳 y鈦�. Calculating this gives us 4 脳 9 = 36, and 36 脳 5 = 180, so the LCD is 180y鈦�. Understanding how to properly combine these factors ensures that you can always find the correct LCD, providing a foundational skill for working with fractions. This approach not only simplifies adding and subtracting fractions but also helps in understanding the relationship between different fractional values.