Problem 1 - Elementary Algebra

Question

For exercises 1-12, use prime factorization to find the least common denominator. $$ \frac{3}{4} ; \frac{5}{6} $$

Step-by-Step Solution

Answer

The least common denominator is 12.

Step 1: Prime Factorization of the Denominators

First, determine the prime factors of each denominator. For 4: 4 = 2 × 2 = 2^2 For 6: 6 = 2 × 3

Step 2: Identify the Highest Powers of All Prime Factors

List all the prime factors found and use the highest power of each. Prime factors: 2, 3 Highest power of 2: 2^2 Highest power of 3: 3

Step 3: Calculate the Least Common Denominator (LCD)

Multiply the highest powers of all prime factors to find the least common denominator (LCD). LCD = 2^2 × 3 = 4 × 3 = 12

Key Concepts

Prime Factorization

To fully understand topics like least common denominator, we should start with prime factorization. Prime factorization breaks down a number into its smallest prime number factors.

For example, in our given exercise, we need the prime factorization of 4 and 6:

• 4 can be factorized into 2 × 2, which we write as 2^2
• 6 can be factorized into 2 × 3, which we write as 2^1 × 3^1

Prime numbers are numbers that can only be divided by 1 and themselves, like 2, 3, 5, and 7.
Knowing how to do prime factorization is crucial for finding the least common multiple (LCM) and the least common denominator (LCD).

Running into numbers like 8 or 12? Just break them down using their smallest prime numbers until you cannot divide any further!

Fractions

Fractions are expressions that tell us how many parts of a whole we have. They are written like this: numerator/denominator.

In our exercise, the fractions are 3/4 and 5/6. Here:

• 3 and 5 are the numerators
• 4 and 6 are the denominators

Sometimes, to make operations easier, you'll need to work with fractions that have the same denominator. This is why finding the least common denominator (LCD) is so important. It simplifies addition and subtraction of fractions.

Without the same denominator, combining fractions becomes tricky and confusing.
Understanding fractions well is key to mastering higher level math concepts. Keep practicing with different fractions to become more comfortable.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. In our exercise, we find the LCM of the denominators (4 and 6) to determine the least common denominator (LCD) for the fractions.

Here’s how it works:

• First, list the prime factorization of each denominator.
• Identify the highest power of each prime factor.
• Multiply these highest powers to find the LCM.

So for 4 (which is 2^2) and 6 (which is 2 × 3), the LCM involves:
Highest power of 2 is 2^2 and the highest power of 3 is 3.
Multiply them to get the LCM, which is 2^2 × 3 = 4 × 3 = 12.
Hence, 12 is our least common denominator (LCD) for the fractions 3/4 and 5/6.
Understanding the concept of LCM makes working with fractions much easier and effective.