Problem 14 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{2 y^{2}}{8 u} $$

Step-by-Step Solution

Answer

Answer: \(\frac{y^2}{4u}\)

Step 1: Identify the numerator and the denominator

In the fraction, the numerator is \(2y^2\) and the denominator is \(8u\).

Step 2: Find the GCD of the coefficients of the numerator and denominator

The coefficients of the numerator and denominator are 2 and 8 respectively. The GCD of 2 and 8 is 2.

Step 3: Divide the numerator and denominator by the GCD

Now, divide both the numerator and the denominator by 2. $$ \frac{2y^2}{8u} \div \frac{2}{2} = \frac{2y^2}{2} \cdot \frac{1}{8u/2} $$

Step 4: Simplify the expression

After simplifying, we have: $$ \frac{1y^2}{4u} $$ So, the rational expression reduced to its lowest terms is \(\frac{y^2}{4u}\).

Key Concepts

Numerator and Denominator

In any rational expression, the numerator is the top part, and the denominator is the bottom part. They are often represented in a fraction format. Understanding the roles of the numerator and the denominator is essential for simplifying fractions.

• The **numerator** tells us how many parts we have. For example, in \( \frac{2y^2}{8u} \), the numerator is \( 2y^2 \).
• The **denominator** tells us into how many parts the whole is divided. In our example, the denominator is \( 8u \).

When you simplify a rational expression, you look to make the relationship between the numerator and the denominator as simple as possible. The goal is to keep the expression equivalent to the original.

Greatest Common Divisor (GCD)

The greatest common divisor, or GCD, is a key concept for reducing fractions. It represents the largest number that divides both the numerator and the denominator without leaving a remainder. To simplify a fraction like \( \frac{2y^2}{8u} \), follow these steps:

• Identify the coefficients from the numerator and denominator. Here, they are 2 and 8.
• Determine the GCD of these numbers. The GCD of 2 and 8 is 2, because 2 is the largest number that can divide into both 2 and 8 evenly.

Using the GCD helps to simplify the overall fraction, allowing you to divide both the numerator and denominator by the GCD, making the fraction easier to interpret and compare to others.

Simplifying Fractions

Simplifying fractions is a step-by-step process that involves reducing the numerator and the denominator by their greatest common divisor. This ensures that the fraction is in its simplest form, meaning no further reduction is possible. Here's how to simplify \( \frac{2y^2}{8u} \):

• Divide both the numerator and the denominator by their GCD, which in this case, is 2.
• Perform the division: \( \frac{2y^2}{2} = y^2 \) and \( \frac{8u}{2} = 4u \).
• Rewrite the expression as \( \frac{y^2}{4u} \).

By consistently using this method, you ensure the expression is in its lowest terms, making it simpler and often easier to work with in larger equations or contexts.