Problem 29 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{3}{8 x}+\frac{7}{10 x} $$

Step-by-Step Solution

Answer

\( \frac{43}{40x} \) is the simplified form.

Step 1: Identify the Common Denominator

The expressions given are \( \frac{3}{8x} + \frac{7}{10x} \). The denominators are \( 8x \) and \( 10x \). To perform the addition, we need a common denominator. The least common denominator (LCD) of \( 8x \) and \( 10x \) is \( 40x \).

Step 2: Rewrite Each Fraction with the Common Denominator

To rewrite \( \frac{3}{8x} \) with the denominator \( 40x \), multiply both the numerator and denominator by 5: \( \frac{3 \times 5}{8x \times 5} = \frac{15}{40x} \).Similarly, to rewrite \( \frac{7}{10x} \) with the denominator \( 40x \), multiply both the numerator and denominator by 4: \( \frac{7 \times 4}{10x \times 4} = \frac{28}{40x} \).

Step 3: Add the Numerators

Now that both fractions have a common denominator, add the numerators: \( \frac{15}{40x} + \frac{28}{40x} = \frac{15 + 28}{40x} = \frac{43}{40x} \).

Step 4: Simplify if Possible

Check if \( \frac{43}{40x} \) can be simplified further. Since 43 is a prime number and does not divide evenly into 40 or any part of \( 40x \), \( \frac{43}{40x} \) is already in its simplest form.

Key Concepts

Least Common Denominator

When dealing with the addition or subtraction of fractions, like in the original exercise, obtaining a Least Common Denominator (LCD) is crucial. The LCD is the smallest expression that is a multiple of each of the denominators involved. It allows you to rewrite fractions with different denominators so that you can combine them.
To find the LCD for rational expressions, first identify the factors in each denominator. For example, in the fractions \( \frac{3}{8x} \) and \( \frac{7}{10x} \):

• The prime factors of 8 are 2 and 4, while for 10, they are 2 and 5.
• Since both expressions also contain an 'x' in the denominator, ensure to include 'x' in the LCD.

Combine these factors at their highest powers to form the LCD. Here, \( 40x \) is obtained because:

• The greatest power of 2 is found in 8 (\(2^3\)).
• Both 4 from 8 and 5 from 10 are included (\(4 \times 5 = 20\)).
• Include 'x' from the variable part.

Thus, the LCD of 8x and 10x is \( 40x \). This method ensures you are working with the smallest possible denominator so that your work is efficient and accurate.

Addition of Fractions

Once you have found the Least Common Denominator (LCD), the next step is the actual addition of fractions. Consider the original fractions \( \frac{3}{8x} \) and \( \frac{7}{10x} \). After determining that the LCD is \( 40x \), you can proceed:

• For \( \frac{3}{8x} \): multiply both the numerator and denominator by 5 to get \( \frac{15}{40x} \).
• For \( \frac{7}{10x} \): multiply both the numerator and denominator by 4 to get \( \frac{28}{40x} \).

The purpose of these operations is to ensure that both fractions now share a common denominator, making them easy to add.
Now, you simply add the numerators while keeping the denominator the same:

• Combine \( \frac{15}{40x} \) and \( \frac{28}{40x} \) to get \( \frac{15 + 28}{40x} \), which simplifies to \( \frac{43}{40x} \).

This operation would not be possible without converting to the common denominator. Remember, the key is to only add or subtract numerators when denominators are identical.

Simplification

Simplification is the final touch in operations with rational expressions. It makes your result tidy and easy to interpret. Upon adding \( \frac{15}{40x} \) and \( \frac{28}{40x} \), we got \( \frac{43}{40x} \); the next question is whether this fraction can be simplified further.
To simplify, check if the numerator and the denominator share any common factors. Here, the numerator is 43 and the denominator is 40x:

• Note that 43 is a prime number.
• Prime numbers do not have divisors other than 1 and themselves.
• The number 40 and the variable 'x' are not divisible by 43.

Thus, \( \frac{43}{40x} \) is already in its simplest form. Simplification ensures that the answer is as straightforward as possible, avoiding unnecessarily large numbers or complexities.