Problem 9 - Elementary Algebra

Question

For exercises 1-12, use prime factorization to find the least common denominator. $$ \frac{1}{50 x^{2} y} ; \frac{1}{35 x y^{3} z} $$

Step-by-Step Solution

Answer

The least common denominator is \( 350x^2y^3z \).

Step 1 - Prime Factorization of Each Denominator

Find the prime factorization of each denominator separately. For the first fraction: \[ 50x^2y = 2 \times 5^2 \times x^2 \times y\]For the second fraction:\[ 35xy^3z = 5 \times 7 \times x \times y^3 \times z\]

Step 2 - Identify All Unique Factors

List all the unique factors that appear in the factorizations from Step 1. The unique factors are:\[ 2, 5, 7, x, y, z\]

Step 3 - Determine the Highest Powers of Each Factor

For each unique factor, determine the highest power that appears in any of the factorizations. This gives us:\[ 2^1, 5^2, 7^1, x^2, y^3, z^1\]

Step 4 - Calculate the Least Common Denominator (LCD)

Multiply the highest powers of each factor together to find the LCD:\[ \text{LCD} = 2^1 \times 5^2 \times 7^1 \times x^2 \times y^3 \times z^1\]So the LCD is:\[ 2 \times 25 \times 7 \times x^2 \times y^3 \times z = 350x^2y^3z\]

Key Concepts

Prime Factorization

Prime factorization breaks down a number into its prime number components. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. In the context of finding the least common denominator (LCD), we use prime factorization to express each denominator in its simplest form. For example, consider the two denominators from the exercise:

• For 50, the prime factorization is 2 and 5 (since 50 = 2 × 5^2).
• For 35, the prime factorization is 5 and 7 (since 35 = 5 × 7).

Breaking down the denominators this way makes it easier to compare and combine the factors to find a common multiple. Prime factorization is a crucial first step in the process of finding the LCD.

Unique Factors

After breaking down each denominator to its prime factors, the next step is to identify all the unique factors. Unique factors are the individual prime numbers or variables that appear in the factorizations of any of the given denominators. For example, from our exercise:

• The factorization of 50x^2y includes 2, 5, x, and y.
• The factorization of 35xy^3z includes 5, 7, x, y, and z.

Combining these, the unique factors are 2, 5, 7, x, y, and z. Identifying these diverse factors is vital as it helps in ensuring that the LCD encompasses all aspects of the individual denominators.

Highest Powers

Once we have identified all unique factors, the next step is to determine the highest powers of each unique factor that appear in any of the factorizations. This step ensures the LCD can accommodate all the individual denominators. Taking from the exercise:

• For the factor 2, the highest power is 2^1 (appears in 50x^2y).
• For the factor 5, the highest power is 5^2 (appears in 50x^2y).
• For the factor 7, the highest power is 7^1 (appears in 35xy^3z).
• For the factor x, the highest power is x^2 (appears in 50x^2y).
• For the factor y, the highest power is y^3 (appears in 35xy^3z).
• For the factor z, the highest power is z^1 (appears in 35xy^3z).

By multiplying these highest powers together, we get the LCD. This step ensures that every denominator can be evenly divided into the common denominator, enabling the fractions to be easily compared or added together.