Problem 148 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{24 a}{32 b^{2}} $$

Step-by-Step Solution

Answer

\(\frac{3a}{4b^{2}}\)

Step 1 - Find the Greatest Common Divisor (GCD)

Identify the GCD of the coefficients. In this case, the GCD of 24 and 32 is 8.

Step 2 - Simplify the Coefficients

Divide both the numerator and the denominator by the GCD. \(\frac{24}{8} = 3\) and \(\frac{32}{8} = 4\). So the fraction becomes \(\frac{3a}{4b^{2}}\).

Step 3 - Check If Further Simplification is Needed

Check if any factors in the variables can be further simplified. In this case, \(a\) and \(b^2\) cannot be further simplified.

Key Concepts

Greatest Common Divisor (GCD)

When simplifying algebraic fractions, finding the Greatest Common Divisor (GCD) is a crucial first step. The GCD is the largest number that can evenly divide both the numerator and the denominator of a fraction. In our example, we have the fraction \(\frac{24a}{32b^2}\). The coefficients are 24 and 32. To find the GCD of 24 and 32, we list the factors of each:

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 32: 1, 2, 4, 8, 16, 32

The largest common factor is 8. Therefore, the GCD of 24 and 32 is 8. This step helps us to simplify the fraction efficiently by reducing it to its simplest form.

Simplification

Simplification is the process of making an algebraic fraction as simple as possible. After finding the GCD, we use it to divide both the numerator and the denominator. This step helps to reduce the fraction without changing its value. In our example, after identifying that the GCD of 24 and 32 is 8, we divide both the numerator and the denominator by 8:

• \(\frac{24}{8} = 3\)
• \(\frac{32}{8} = 4\)

Therefore, the simplified form of \(\frac{24a}{32b^2}\) becomes \(\frac{3a}{4b^2}\). Simplification also includes checking if any variable terms can be further simplified. In this case, neither \(a\) nor \(b^2\) can be reduced further, so \(\frac{3a}{4b^2}\) is the simplest form.

Fractions

Understanding fractions is fundamental when dealing with algebraic expressions. A fraction consists of a numerator (the top part) and a denominator (the bottom part). In algebraic fractions, both the numerator and denominator can include variables and constants. For example, in the fraction \(\frac{24a}{32b^2}\), the numerator is 24a and the denominator is 32b^2.
Simplifying algebraic fractions involves:

• Identifying common factors in the numerator and denominator
• Dividing both parts by those common factors (the GCD)

This process ensures that the fraction is in its simplest form. By making the fraction easier to understand, it becomes more manageable to work with in broader algebraic problems. Always make sure to check for any possible further simplifications, even after using the GCD.