Problem 9 - Elementary Algebra

Question

For the following problems, reduce each rational expression to lowest terms. $$ \frac{8}{4 a-16} $$

Step-by-Step Solution

Answer

Answer: $$\frac{1}{2(a-4)}$$.

Step 1: Identify the Greatest Common Divisor (GCD)

First, we have to find the GCD of the numerator and denominator. In this case, the numerator is 8 and the denominator is $$4a - 16$$. The GCD of 8 and 16 is 8.

Step 2: Factor out the GCD in the numerator and denominator

Now, we can factor out the GCD from both the numerator and the denominator. This gives us: $$ \frac{8}{4a-16} = \frac{8}{4(a-4)} $$

Step 3: Cancel out the common factor of 8

Since both the numerator and the denominator have a common factor of 8, we can cancel them out, which leaves us with the simplified rational expression: $$ \frac{8}{4(a-4)} = \frac{8}{4 \cdot 2(a-4)} = \frac{1}{2(a-4)} $$ The simplified rational expression is $$\frac{1}{2(a-4)}$$.

Key Concepts

Greatest Common Divisor

When handling rational expressions, finding the greatest common divisor (GCD) is a crucial step. The GCD is the largest number that can evenly divide both the numerator and the denominator. Identifying this can simplify the expression significantly. For example, in the expression \( \frac{8}{4a-16} \), we seek the GCD between 8, the constant in the numerator, and the terms 4 and 16 in the polynomial denominator.

• To find the GCD of 8 and 16, list all their divisors.
• The divisors of 8 are 1, 2, 4, and 8.
• The divisors of 16 are 1, 2, 4, 8, and 16.
• The largest common number in both lists is 8.

Recognizing and understanding the GCD allows us to progress to factorization more effectively.

Factorization

Factorization involves expressing a mathematical expression as a product of its factors. In rational expressions, this step allows us to break down the denominator or numerator to identify and cancel common terms. With the expression \( \frac{8}{4a-16} \), once we've identified that the GCD is 8, we can concentrate on factoring the denominator.
We break down the denominator by taking out the common factor:

• The expression \( 4a - 16 \) includes terms that share the common factor of 4.
• By factoring out 4, we rewrite the expression as \( 4(a-4) \).
• This makes the rational expression \( \frac{8}{4(a-4)} \).

Factoring helps in simplifying expressions by revealing hidden common factors that can be canceled.

Canceling Factors

The process of simplifying rational expressions often involves canceling common factors in the numerator and the denominator. This step is possible only after factorization, making it essential for reducing expressions to their lowest terms. In our example, \( \frac{8}{4(a-4)} \), both the numerator and the newly factored denominator have a common factor of 8.

• To simplify, divide both the numerator and the denominator by 8.
• This operation removes the 8, resulting in the expression \( \frac{1}{2(a-4)} \).

Remember to only cancel factors that are identical and shared between the numerator and the denominator. This ensures the expression remains equivalent to the original.