Problem 44 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{7}{2 x^{2}}+\frac{1}{6 x^{3}} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions: $$\frac{7}{2x^2} + \frac{1}{6x^3}$$ Answer: The sum of the given rational expressions is $$\frac{21x+1}{6x^3}$$.

Step 1: Find the least common denominator (LCD)

In order to add the rational expressions, we need a common denominator. We start by finding the least common multiple (LCM) of the two denominators: \(2x^2\) and \(6x^3\). The LCM is the smallest expression that both denominators evenly divide into. First, find the prime factors of each number: 2: 2 6: 2 * 3 Now, for the LCM, we need the highest powers of all prime factors that appear in denominators: LCM numbers: 2^1, 3^1 -> 2 * 3 = 6 For the variables, we need the highest exponent value for each variable in the denominators: LCM variables: \(x^3\) Now, we can combine the LCM of numbers and variables: LCD = 6x^3

Step 2: Express each term using the common denominator

Now that we have the common denominator (6x^3), we can rewrite each term by multiplying its numerator and denominator by the appropriate expression to make sure the denominators match the LCD: $$ \frac{7}{2 x^{2}} * \frac{3x}{3x} = \frac{21x}{6 x^{3}} $$ $$ \frac{1}{6 x^{3}} * \frac{1}{1} = \frac{1}{6 x^{3}} $$ So, the expression becomes: $$ \frac{21x}{6x^3}+\frac{1}{6x^3} $$

Step 3: Add the rational expressions

Now that both terms have the same denominator (6x^3), we can add the numerators of these fractions: $$ \frac{21x+1}{6x^3} $$

Step 4: Simplifying the result

Check if there are any common factors for the numerator and the denominator that can be simplified. In this case, there are no common factors, so the final answer is: $$ \frac{21x+1}{6x^3} $$

Key Concepts

Least Common Denominator

When working with rational expressions, having a common denominator is crucial for operations like addition or subtraction. This is where the concept of the Least Common Denominator (LCD) comes into play. The LCD is essentially the least common multiple (LCM) of the denominators involved.

To find it, we begin by determining the LCM of both the numerical coefficients and variables separately. Numerical coefficients are numbers without variables, such as 2 and 6 in the problem. Follow these general steps:

• Find all prime factors of each number. For example, 2 is just 2, but 6 breaks down into 2 and 3.
• Identify the highest power for these prime factors across the denominators. In our example, we have 2¹ and 3¹ for the numbers, giving us a product of 6.
• For variables, identify the highest power present in any of the denominators. If you have denominators like \( x^2 \) and \( x^3 \), you select \( x^3 \).

Combine the LCM of both numbers and variables to find the LCD: in this problem, it's \( 6x^3 \). This forms the basis for expressing all terms to a common denominator.

Adding Fractions

Once the least common denominator has been established, the next step is to rewrite each fraction so that they share this common denominator. This alignment allows us to effectively add the fractions together.

With the common denominator \(6x^3\), make sure each fraction corresponds to this denominator:

• Multiply the numerator and denominator of the first fraction \(\frac{7}{2x^2}\) by \(3x\), to match the LCD. It becomes \(\frac{21x}{6x^3}\).
• The second fraction \(\frac{1}{6x^3}\) already has the common denominator so it remains unchanged.

Thus, the expression becomes \(\frac{21x}{6x^3} + \frac{1}{6x^3}\). Ensure consistency in the denominators before proceeding to the addition stage.

Simplifying Fractions

With both fractions rewritten with a shared denominator, it's time to combine them. The process of adding fractions becomes simple when there's a common denominator.

Here's what to do:

• Combine the numerators while retaining the common denominator, leading to \(\frac{21x + 1}{6x^3}\).

Next, check if the resulting fraction can be simplified. To simplify, look for common factors in both the numerator and the denominator. If found, you can reduce the fraction.

For \(\frac{21x+1}{6x^3}\), there are no common factors between the numerator and denominator, so it remains in its simplest form. Simplifying fractions ensures we have the most reduced and manageable form of the expression.

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. It's a key step in finding the least common denominator for rational expressions.

To break down a number into its prime factors, consider the nature of the number itself. Any number can be represented as a combination of prime factors.

• Take 2, a prime number in itself, so its prime factorization is just 2.
• For the number 6, you multiply the primes 2 and 3 to get 6.

By decomposing each number into prime factors, particularly for numerical coefficients in denominators, you lay the groundwork for calculating the least common denominator.

Understanding prime factorization not only aids in resolving issues of least common denominators but also ensures accuracy in arithmetic involving fractions.