Problem 149 - Intermediate Algebra

Question

In the following exercises, simplify. $$ -\frac{210 a^{2}}{110 b^{2}} $$

Step-by-Step Solution

Answer

-\frac{21 a^{2}}{11 b^{2}}

Step 1: Identify Common Factors

First, let's identify the greatest common divisor (GCD) for the numerator and the denominator. The given fraction is \(-\frac{210 a^{2}}{110 b^{2}}\). We need to simplify this by dividing both the numerator and the denominator by their GCD.

Step 2: Find the GCD of the Numbers

The numbers in the fraction are 210 and 110. We need to find the greatest common divisor of these two numbers. The GCD of 210 and 110 is 10.

Step 3: Simplify the Fraction by Dividing by the GCD

Divide both the numerator and the denominator by 10. This gives us: \[ -\frac{210 \, a^{2}}{110 \, b^{2}} = -\frac{210 \div 10 \, a^{2}}{110 \div 10 \, b^{2}} = -\frac{21 \, a^{2}}{11 \, b^{2}} \]

Step 4: Final Simplified Form

The fraction is now in its simplest form, which is \( -\frac{21 a^{2}}{11 b^{2}} \). There are no more common factors between the numerator and the denominator.

Key Concepts

greatest common divisor

To simplify an algebraic fraction, the first step is to find the greatest common divisor (GCD). The GCD of two numbers is the largest positive integer that divides both of them without leaving a remainder.
In our example, we need to find the GCD for 210 and 110. To do this, you can use the prime factorization method:
- The prime factors of 210 are 2, 3, 5, and 7.- The prime factors of 110 are 2, 5, and 11.
The common prime factors are 2 and 5. Therefore, the GCD of 210 and 110 is the product of these common factors, which is 10.
Once you have the GCD, you can divide both the numerator and the denominator of your fraction by this number to simplify it.

numerator and denominator

In algebraic fractions, the numerator is the top part, and the denominator is the bottom part. For the expression \(-\frac{210 \ a^{2}}{110 \ b^{2}}\), the numerator is 210 \ a^{2}\, and the denominator is 110 \ b^{2}\.
To simplify a fraction, you need to work on both the numerator and the denominator concurrently.
By identifying common factors in the numerator and the denominator, we can divide both by their GCD. This process helps reduce the fraction to its simplest form.
In our example, once we found the GCD as 10, we divided both the numerator and the denominator by 10, resulting in \(-\frac{21 \ a^{2}}{11 \ b^{2}}\).

simplification

Simplifying algebraic fractions involves reducing them to their simplest form by eliminating common factors. Here’s how you can simplify:

• Identify the GCD of the numbers in the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

In our example, we started with the fraction \(-\frac{210 \ a^{2}}{110 \ b^{2}}\).
By identifying the GCD (which is 10) and dividing both parts by it, we reduced the fraction to \(-\frac{21 \ a^{2}}{11 \ b^{2}}\).
This fraction is now in its simplest form because there are no more common factors between the numerator and the denominator. This process makes algebraic fractions easier to work with and understand.