Problem 28 - Elementary and Intermediate Algebra

Question

Subtract and simplify the result, if possible. \(\frac{c+7}{4 c^{4}}-\frac{3}{4 c^{4}}\)

Step-by-Step Solution

Answer

\(\frac{c+4}{4c^4}\)

Step 1: Identify like terms

Observe the expression \( \frac{c+7}{4 c^{4}}-\frac{3}{4 c^{4}} \). Notice that both fractions have the same denominator \( 4c^4 \). This means they are like terms and can be combined directly.

Step 2: Subtract the numerators

Since the denominators are the same, subtract the numerators: \( (c+7) - 3 \). This becomes \( c + 7 - 3 \).

Step 3: Simplify the numerator

Simplify the expression \( c + 7 - 3 \). Combine the constant terms to get \( c + 4 \).

Step 4: Write the result as a single fraction

The expression now becomes \( \frac{c+4}{4c^4} \), since we have combined the numerators over the common denominator.

Key Concepts

Subtraction in Algebra

In algebra, subtraction can be tricky because it's important to carefully follow the rules for combining like terms. When dealing with expressions that involve subtraction, you must pay attention to the numerators and denominators, especially when dealing with fractions. In our exercise, we subtracted fractions with a common denominator. This simplifies the process, as you only need to subtract the numerators directly. Remember, when subtracting algebraic expressions, handle each term of the numerators separately.

• Ensure the denominators are the same before subtraction.
• Line up the numerators and perform the subtraction.
• Check if the result can be simplified further.

Carefully organizing each step helps in achieving the correct result in algebraic subtraction processes.

Like Terms

Like terms are terms whose variables and their exponents are the same. In our exercise, both fractions, \( \frac{c+7}{4c^4} \) and \( \frac{3}{4c^4} \) shared a common denominator of \( 4c^4 \), meaning these terms can be directly combined. Aligning like terms is an invaluable step when simplifying complex expressions. Here’s why like terms matter:

• They allow you to combine expressions more efficiently.
• Combining like terms streamlines simplifying the overall expression.
• Reduces complexity by minimizing the number of distinct terms present.

Always ensure to spot like terms in an algebraic expression to simplify it easily and avoid unnecessary complexity.

Simplifying Fractions

Simplifying fractions in algebra involves reducing them to their simplest form. This process can make complex algebraic expressions much easier to work with. Simplifying involves combining like terms in both the numerator and the denominator, and potentially canceling common factors. In our exercise, after subtracting, the fraction \( \frac{c+4}{4c^4} \) is already in its simplest form, as there are no common factors between \( c+4 \) and \( 4c^4 \) that need canceling.

• Look for any common factors between the numerator and denominator.
• Divide both the numerator and denominator by these factors if any exist.
• Simplification helps in making calculation steps clearer and avoiding errors.

It is essential to always simplify fractions to ease computation and enhance understanding of the algebraic expressions you are working with.