Problem 30 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{12 x}{16 x^{7}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3}{4x^6} \).

Step 1: Identify Common Factors

Look at the numerator, which is \(12x\), and the denominator, which is \(16x^7\). Both terms contain \(x\), and the numbers 12 and 16 are divisible by a common factor.

Step 2: Greatest Common Factor (GCF) of the Coefficients

The greatest common factor of 12 and 16 is 4. So, we can factor out 4 from both the numerator and the denominator.

Step 3: Divide the Coefficients by the GCF

Divide the coefficients: \( \frac{12}{4} = 3 \) and \( \frac{16}{4} = 4 \). This simplifies the expression to \( \frac{3x}{4x^7} \).

Step 4: Simplify the Variables

Apply the rule of division for exponents: \( \frac{x}{x^7} = x^{1-7} = x^{-6} \). Therefore, the expression becomes \( \frac{3}{4x^6} \).

Step 5: Final Simplified Form

The rational expression simplifies to \( \frac{3}{4x^6} \). There are no further common factors to simplify.

Key Concepts

Understanding the Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is crucial in simplifying expressions, especially when dealing with coefficients. The GCF is the largest number that divides two or more integers without leaving a remainder. In the context of our exercise, we're focusing on the numbers 12 and 16.

• For 12, the factors are: 1, 2, 3, 4, 6, 12
• For 16, the factors are: 1, 2, 4, 8, 16

The largest number that appears in both lists is 4. By dividing both the numerator and the denominator by this GCF, we simplify the rational expression. Remember, finding the GCF helps make the numbers smaller and the expression easier to work with.

Mastering the Division of Exponents

When simplifying expressions that involve exponents, knowing how to divide exponents is a handy tool. The rule is quite simple: when you divide like bases, you subtract the exponents. In this exercise, the variable parts involve division of exponents, specifically with the base 'x'.Here's how it works:

• Start with the expression: \( \frac{x}{x^7} \)
• Apply the division rule: subtract the exponents, giving you \( x^{1-7} \)
• This results in \( x^{-6} \)

A negative exponent indicates that the base is on the wrong side of the fraction. So, \( x^{-6} \) means \( \frac{1}{x^6} \). Understanding this division rule helps manage and simplify complex rational expressions easily.

Navigating Rational Expressions

Rational expressions are similar to fractions but involve polynomials instead of integers. This exercise deals with simplifying a specific type of rational expression. Simplification of rational expressions often involves factorization and reducing fractions. Here's a quick guide to handling these mathematical problems:

• **Identify Sounds**: Check for common factors in both the numerator and denominator.
• **GCF**: Use the greatest common factor to reduce coefficients.
• **Divide Exponents**: Apply the exponent rules to simplify variable factors.

The final aim is to express the rational expression in its simplest form, just like reducing a fraction. The expression \( \frac{12x}{16x^7} \) gets simplified to \( \frac{3}{4x^6} \) after applying these steps, embodying the essence of simplification.