Problem 34 - Elementary and Intermediate Algebra

Question

Simplify each rational expression. See Example 3 . $$\frac{12 x}{16 x^{7}}$$

Step-by-Step Solution

Answer

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

Step 1: Identify Common Factors

The expression is \( \frac{12x}{16x^7} \). First, separate the coefficients and the variables to identify common factors. The coefficient in the numerator is 12 and in the denominator is 16. For the variables, we have \(x\) in the numerator and \(x^7\) in the denominator.

Step 2: Simplify the Coefficients

The coefficients are 12 and 16. Find the greatest common factor (GCF) of 12 and 16, which is 4. Divide both coefficients by this GCF: \( \frac{12}{4} = 3 \) and \( \frac{16}{4} = 4 \). Thus, the coefficients simplify to \( \frac{3}{4} \).

Step 3: Simplify the Variables

For the variables, divide the powers of \(x\). In \( \frac{x}{x^7} \), apply the rule \( \frac{x^m}{x^n} = x^{m-n} \), where \( m = 1 \) and \( n = 7 \). Therefore, \( x^{1-7} = x^{-6} \).

Step 4: Combine Simplified Parts

Combine the simplified coefficient and variable parts: \( \frac{3}{4} \cdot x^{-6} \). The final simplified expression is \( \frac{3}{4x^6} \) since \( x^{-6} = \frac{1}{x^6} \).

Key Concepts

Greatest Common Factor

When dealing with rational expressions, such as the one given in the exercise, finding the Greatest Common Factor (GCF) is a crucial step. The GCF is the largest number that divides two or more numbers without leaving a remainder. Identifying and using the GCF helps in simplifying expressions by reducing the coefficients to their simplest form.

For example, in the expression \( \frac{12x}{16x^7} \), the coefficients are 12 and 16. The GCF of these two numbers is 4 because 4 is the largest number that can evenly divide both 12 and 16. By dividing 12 and 16 by their GCF, we simplify them to \( \frac{12}{4} = 3 \) and \( \frac{16}{4} = 4 \) respectively.

• Helps in reducing coefficients in expressions
• Makes large numbers more manageable
• Ensures the expression is in its simplest form

Always remember to find the GCF first before simplifying the rest of the expression. This will make the calculations smoother and correct.

Exponent Rules

Exponent rules play a significant role in simplifying rational expressions that include variables. An exponent indicates how many times a number, known as the base, is multiplied by itself. Specific rules apply when simplifying, multiplying, or dividing terms with exponents.

In our exercise, we encounter the expression \( \frac{x}{x^7} \). To simplify this, we use the exponent division rule: \( \frac{x^m}{x^n} = x^{m-n} \). This means we subtract the exponent in the denominator from the exponent in the numerator. In this case, \( m = 1 \) (since \( x \) is the same as \( x^1 \)) and \( n = 7 \), allowing us to simplify as follows:

\( x^{1 - 7} = x^{-6} \).

• \( a^m \times a^n = a^{m+n} \)
• \( \frac{a^m}{a^n} = a^{m-n} \)
• \( a^{-m} = \frac{1}{a^m} \)

These rules are essential for manipulating and simplifying expressions with exponents. They ensure the final expression is expressed in a clear, concise, and mathematically correct way.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. Simplifying them involves reducing the expression to its simplest form, similar to how we simplify numerical fractions. This process generally includes factoring polynomials and using the greatest common factor along with exponent rules.

In the expression \( \frac{12x}{16x^7} \), we both simplified the coefficients using the GCF and applied exponent rules to the variables. This resulted in streamlining the expression to \( \frac{3}{4x^6} \). While rational expressions might seem complex, they can be made easier by breaking them into parts and solving each in steps.

• Identify and factor the expression completely
• Simplify coefficients and variable parts separately
• Combine the simplified components for the final result

Remember, mastering the art of simplifying rational expressions requires understanding and applying the rules consistently. With practice, you'll find it becomes more intuitive.