Problem 30 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{6 a^{3} b^{12} c^{5}}{-8 a b^{4} c^{9}}$$

Step-by-Step Solution

Answer

\(-\frac{3 a^{2} b^{8}}{4 c^{4}}\)

Step 1: Identify the numerator and the denominator

The given rational expression is \(\frac{6 a^{3} b^{12} c^{5}}{-8 a b^{4} c^{9}}\). Here, the numerator is \(6 a^{3} b^{12} c^{5}\), and the denominator is \(-8 a b^{4} c^{9}\).

Step 2: Factor out the coefficients

Factor the coefficients (numbers) in the numerator and denominator: \(6\) and \(-8\) respectively. The greatest common divisor (GCD) of \(6\) and \(-8\) is \(2\). Factor \(2\) from both: \(\frac{6}{-8} = \frac{6 \times 1}{8 \times -1} = \frac{3}{-4}\).

Step 3: Simplify the variable terms

Divide the variable terms in the numerator by the corresponding terms in the denominator:

Step 3a: Simplify the \(a\)-terms

\(\frac{a^{3}}{a} = a^{3-1} = a^{2}\)

Step 3b: Simplify the \(b\)-terms

\(\frac{b^{12}}{b^{4}} = b^{12-4} = b^{8}\)

Step 3c: Simplify the \(c\)-terms

\(\frac{c^{5}}{c^{9}} = c^{5-9} = c^{-4} = \frac{1}{c^{4}}\).

Step 4: Combine the simplified terms

Combine the results from the previous steps: \(\frac{3 a^{2} b^{8}}{-4 c^{4}}\).

Step 5: Express the final simplified form

The negative sign can be placed in front of the fraction to indicate the expression is negative: \(-\frac{3 a^{2} b^{8}}{4 c^{4}}\).

Key Concepts

Greatest Common Divisor (GCD)

When simplifying rational expressions, finding the Greatest Common Divisor (GCD) of coefficients is crucial. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For instance, in our exercise, we have the numbers 6 and -8. We factor each number into its prime factors: 6 is 2 × 3, and -8 is -2 × 2 × 2. Here, the common factor is 2. So, the GCD of 6 and -8 is 2. By dividing both the numerator and the denominator by 2, we simplify the coefficients in the expression from 6 and -8 to 3 and -4, respectively. This simplification is essential for reducing the rational expression to its simplest form. Remember, always look for the highest possible number that divides both coefficients completely to find the GCD.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms. This means dividing the numerator and the denominator by their Greatest Common Divisor (GCD). In rational expressions, we apply this concept to both numerical and variable terms.
For example, the expression \(\frac{6 a^{3} b^{12} c^{5}}{-8 a b^{4} c^{9}}\) has both coefficients and variable terms. First, handle the numerical part. We already found the GCD of 6 and -8, which is 2. Hence, \(\frac{6}{-8}\) simplifies to \(\frac{3}{-4}\). Next, we simplify the variable terms by using exponent rules, dividing the powers of the same base by subtracting their exponents. This gives us simplified variable terms, transforming the original fraction step by step into a simpler form.
By dividing both the numerator and the denominator by their common factors, we ensure the fraction is in its simplest state. This process not only makes the expression easier to work with but also reveals its true, simplified form.

Exponent Rules

Understanding exponent rules is essential when simplifying fractions with variable terms. These rules help to manage and simplify expressions involving powers of the same base. The key rules include:

• Product of Powers Rule: \(a^{m} \times a^{n} = a^{m+n}\). This rule states that when multiplying like bases, you add the exponents.
• Quotient of Powers Rule: \(a^{m} \div a^{n} = a^{m-n}\). This is crucial for our exercise, as it simplifies the division of terms with the same base.
• Negative Exponent Rule: \(a^{-n} = \frac{1}{a^{n}}\).

Let's break down the exercise using these rules:
- For \(a\) terms: \(\frac{a^{3}}{a} = a^{2}\).
- For \(b\) terms: \(\frac{b^{12}}{b^{4}} = b^{8}\).
- For \(c\) terms which include a negative exponent: \(\frac{c^{5}}{c^{9}} = c^{-4} = \frac{1}{c^{4}}\).
By applying these exponent rules, we can efficiently simplify the variable components in the rational expression. This makes it manageable and reveals the most reduced form, ensuring accurate and concise results.