Problem 10 - Elementary Algebra

Question

For exercises 1-12, use prime factorization to find the least common denominator. $$ \frac{1}{40 x y^{2} z} ; \frac{1}{42 x y z^{3}} $$

Step-by-Step Solution

Answer

The least common denominator is 840xy^2z^3.

Step 1: Prime Factorization of 40

Prime factorize the number 40. You get: 40 = 2^3 * 5

Step 2: Prime Factorization of 42

Prime factorize the number 42. You get: 42 = 2 * 3 * 7

Step 3: List All Variables and Their Powers

Consider the variables and their powers from both denominators:1st fraction: 40xy^2z -> 2^3, 5, x, y^2, z2nd fraction: 42xyz^3 -> 2, 3, 7, x, y, z^3

Step 4: Identify the Highest Powers of Each Prime Factor and Variable

List the highest powers of each unique factor:2^3 (from 40)3 (from 42)5 (from 40)7 (from 42)x (both have x)y^2 (from 40xy^2z)z^3 (from 42xyz^3)

Step 5: Multiply to Get the LCD

Multiply the highest powers of each prime factor and variable to get the least common denominator (LCD):LCD = 2^3 * 3 * 5 * 7 * x * y^2 * z^3 = 840xy^2z^3

Key Concepts

Prime Factorization

Prime factorization is an essential skill in mathematics.
It involves breaking down a number into its prime number building blocks.
Let's look at the example from our exercise. For the number 40, we determine its prime factors by breaking it down step by step: 40 = 2 * 20 = 2 * 2 * 10 = 2 * 2 * 2 * 5, or more compactly, 40 = 2^3 * 5.
Similarly, for 42, we do the same: 42 = 2 * 21 = 2 * 3 * 7.
These prime factors (2, 3, 5, and 7) help us find common denominators for fractions.
Remember, prime factorization is like unlocking the code of a number!

Variables and Powers

Variables and powers can complicate finding a least common denominator (LCD).
In our example, we have variables like x, y, and z in the denominators, each raised to different powers.
The first fraction has 40xy^2z, and the second has 42xyz^3. To find the LCD, we look at the variables: both have x (to the power of 1), y^2 (from the first fraction), and z^3 (from the second).
When listing variables and powers, always take the highest power appearing in any of the fractions.
This ensures that any factor missing in one fraction is accounted for in the least common denominator.

Multiplying Factors

Multiplying the factors involves combining both numbers and variables.
You must consider the highest powers of each prime number and variable. From our exercise, those are: 2^3, 3, 5, 7, x, y^2, and z^3.
Multiply these factors together to find the LCD: \[LCD = 2^3 \times 3 \times 5 \times 7 \times x \times y^2 \times z^3 = 840xy^2z^3\]This product includes all necessary factors from both denominators, ensuring that each fraction’s denominator divides the LCD perfectly.
Multiplying correctly ensures the final denominator can be used to add or subtract fractions efficiently.

Educational Mathematics

Understanding these concepts aids in mastering foundational math skills.
Knowing how to find the least common denominator using prime factorization and variables is critical for advanced mathematics.
This process helps simplify complex fractions, making addition and subtraction straightforward.
For educators, breaking down each step ensures students follow the logic behind the calculations.

• First: Prime Factorize each number.
• Second: List the highest powers of each prime factor and variable from the denominators.
• Finally: Multiply these together to find the least common denominator.

Practice makes perfect in educational mathematics. Using these steps regularly helps reinforce the method, ensuring students can tackle similar problems with confidence.