Problem 16 - Algebra A Combined Function

Question

Find the \(L C D\) for each list of rational expressions. $$ \frac{17 x}{4 y^{5}}, \frac{2}{8 y} $$

Step-by-Step Solution

Answer

The LCD is \(8y^5\).

Step 1: Identify the Denominators

First, observe the denominators of the expressions. The first denominator is \(4y^5\) and the second is \(8y\).

Step 2: Factor the Denominators

Factor each denominator into its prime components. \(4y^5 = 2^2 y^5\) and \(8y = 2^3 y\) are the factored forms of the denominators.

Step 3: Determine the LCD Factors

To find the LCD (Least Common Denominator), take the highest power of each prime factor present in the denominators. For \(2\), the highest power is \(2^3\). For \(y\), the highest power is \(y^5\).

Step 4: Calculate the LCD

Multiply the highest powers of all prime factors to get the LCD: \(2^3 \times y^5 = 8y^5\). Thus, the LCD is \(8y^5\).

Key Concepts

Rational Expressions

Rational expressions are fractions in which both the numerator and the denominator are polynomials. They are similar to rational numbers, which are fractions where both the numerator and the denominator are integers.
A rational expression can look like \( \frac{17x}{4y^5} \) or \( \frac{2}{8y} \). In these expressions, the numerator and the denominator can both be composed of coefficients (like 17 or 2) and variables (like \( x \) or \( y \)).

• The key is to understand that simplifying or finding common denominators with rational expressions often involves polynomial expressions, instead of simple integers.
• When working with rational expressions, we treat the polynomial in the denominator as a single entity throughout calculations.

Understanding rational expressions is essential as you learn to manipulate, simplify, and solve problems involving them.

Factoring

Factoring is the process of breaking down an expression into simpler components, or 'factors', that when multiplied together give the original expression. Factoring is crucial in simplifying rational expressions and finding the Least Common Denominator (LCD).
In the context of our exercise, consider the denominators \( 4y^5 \) and \( 8y \).

• The factors of \( 4y^5 \) are \( 2^2 \) and \( y^5 \).
• The factors of \( 8y \) are \( 2^3 \) and \( y \).

Factoring makes it easier to compare the denominators by seeing each as a product of its simplest elements. This way, you can easily identify the highest powers of each factor necessary for calculating the LCD.

Prime Factors

Prime factors are the building blocks of numbers and expressions in mathematics, derived by factoring them down to their most basic components – prime numbers.
To find the prime factors of a denominator, you must break it down completely into its prime components. In the given exercise:

• \( 4y^5 \)'s prime factors are \( 2^2 \) and \( y^5 \).
• \( 8y \)'s prime factors are \( 2^3 \) and \( y \).

Understanding which prime factors are present and their highest powers helps you find the Least Common Denominator. You'll know exactly which factors and exponents to multiply together to cover all variants from each expression in the problem.

Denominators

Denominators play a significant role in rational expressions as they determine the base upon which the expression operates. In any pair of rational expressions, such as in our exercise, the denominators must align to perform operations like addition or subtraction.
In this particular exercise, the denominators are \( 4y^5 \) and \( 8y \), and the goal is to find a common ground, or LCD, for both expressions.
The process involves:

• Identifying the denominators you are working with.
• Factoring them into their prime factors.
• Determining the highest power of each factor needed.

Finally, combining these to find the LCD as \( 8y^5 \). With the denominator issues sorted, one can confidently work on the expressions individually or together as needed.