Problem 41 - Elementary and Intermediate Algebra

Question

Simplify. See Example 3. $$ \frac{6 x^{4}}{4 x^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{3x^2}{2} \).

Step 1: Understand the Problem

We need to simplify the expression \( \frac{6x^4}{4x^2} \). This involves reducing the expression by cancelling out common terms between the numerator and the denominator.

Step 2: Simplify the Coefficient

Look at the coefficients (numbers) in the fraction: \( \frac{6}{4} \). Simplify this fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2. Thus, \( \frac{6}{4} \) simplifies to \( \frac{3}{2} \).

Step 3: Simplify the Variables

For the variable part, \( \frac{x^4}{x^2} \), apply the rule of exponents: \( \frac{x^m}{x^n} = x^{m-n} \). Here, \( m=4 \) and \( n=2 \), so this becomes \( x^{4-2} = x^2 \).

Step 4: Combine the Simplified Parts

Combine the simplified coefficient and the simplified variable: \( \frac{3}{2}x^2 \). Thus, the simplified expression is \( \frac{3x^2}{2} \).

Key Concepts

Exponent Rules

Exponent rules are fundamental when dealing with expressions involving powers of the same base. Here, we specifically focus on the division rule of exponents. When you divide two expressions with the same base, like in the fraction \( \frac{x^4}{x^2} \), the rule \( \frac{x^m}{x^n} = x^{m-n} \) comes into play. This rule tells us to subtract the exponent of the denominator from the exponent of the numerator. In our exercise, the numerator's exponent is 4, and the denominator's exponent is 2. Thus, you calculate \( x^{4-2} = x^2 \). This process drastically reduces the power expression while maintaining its mathematical integrity. Remember, this rule only applies when you're dealing with the same base in both parts of the fraction.

Greatest Common Divisor

Finding the greatest common divisor (GCD) is a key step in simplifying fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. In our example, we had the fraction \( \frac{6}{4} \). To simplify, we must determine the GCD of 6 and 4.

• Number 6: Its divisors are 1, 2, 3, and 6.
• Number 4: Its divisors are 1, 2, and 4.
• The greatest common divisor here is 2, as it is the largest number that fits both lists.

By dividing both the numerator and the denominator by 2, you achieve the simplified fraction \( \frac{3}{2} \). This step is crucial for reducing fractions to their simplest form.

Fraction Simplification

Fraction simplification involves a couple of neat tricks to make the expression easier to work with. The ultimate aim is to reduce a fraction to its simplest form, making it less complex and more manageable. First, take note of both the numeric (coeficients) and variable parts of the fraction. Each should be simplified separately for clarity.

• For the numerical part, we use the GCD, as discussed, to reduce the fraction. For \( \frac{6}{4} \), this gives us \( \frac{3}{2} \).
• For the variable part, like \( \frac{x^4}{x^2} \), applying exponent rules, we simplify to \( x^2 \).

Once both parts are individually simplified, they can be combined to achieve the final expression. In this case, it results in \( \frac{3x^2}{2} \). The process of simplifying ensures that fractions are more straightforward to interpret and use for further calculations. Always remember, a well-simplified fraction is the foundation of efficient problem-solving in algebra!