Problem 63 - Algebra 2

Question

Add or subtract. Simplify where possible. $$ \frac{7}{2 c}-\frac{2}{c^{2}} $$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{7c-2}{2c^2}\).

Step 1: Identify the Least Common Denominator (LCD)

The denominators are \(2c\) and \(c^2\). The least common denominator is the least common multiple (LCM) of these two denominators which is \(2c^2\).

Step 2: Rewrite the fractions using the LCD

Rewrite each of the fractions with the new denominator by multiplying the top and bottom of the first fraction by \(c\) and leaving the second fraction as is. We get \(\frac{7c}{2c^2}-\frac{2}{2c^2}\).

Step 3: Perform the subtraction

Now, both fractions share the same denominator, the subtraction can be directly performed: \(\frac{7c-2}{2c^2}\).

Key Concepts

Least Common Denominator

When working with algebraic fractions, combining fractions requires a shared denominator. This shared denominator is known as the Least Common Denominator (LCD). It's essential for ensuring a smooth and error-free calculation process.

Finding the LCD involves determining the smallest multiple that the denominators share. For example, if the denominators are \(2c\) and \(c^2\), calculate the least common multiple (LCM) of these expressions. The LCM of \(2c\) and \(c^2\) involves taking the highest power of each distinct variable and constant:

• The highest power of \(c\) present is \(c^2\).
• The constant term "2" combines as it appears, since it is present in the \(2c\).

Thus, the LCD is \(2c^2\). By rewriting the fractions with this common denominator, we can easily add or subtract them without any hassle.

Subtracting Fractions

Subtracting algebraic fractions is quite similar to subtracting numerical fractions. After establishing a common denominator, the numerators are subtracted while the denominator remains unchanged.

In the example \(\frac{7}{2c}-\frac{2}{c^2}\), once we've established our LCD as \(2c^2\), we rewrite both fractions so they have this common denominator:

• Multiply \(\frac{7}{2c}\) by \(\frac{c}{c}\) to become \(\frac{7c}{2c^2}\).
• The second fraction, \(\frac{2}{c^2}\), is then rewritten directly as \(\frac{2}{2c^2}\) since the denominator already fits the LCD.

With the fractions now compatible, subtract the numerators directly: \(7c - 2\). Place the result over the shared denominator to get \(\frac{7c-2}{2c^2}\). Ensure the subtraction is performed carefully to avoid mistakes.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This involves factoring both the numerator and the denominator and then cancelling common factors.

In the fraction \(\frac{7c-2}{2c^2}\), we would explore if there are common factors in the numerator and denominator that can cancel out. At times, the expression is already in its simplest form and no further reduction is possible. If available, perform these steps:

• Look for common factors between the numerator \(7c - 2\) and the denominator \(2c^2\).
• Check if any term or coefficient in the numerator and denominator can be divided by the same number or expression.

In this specific example, there are no common factors that allow further simplification, thus the expression \(\frac{7c-2}{2c^2}\) remains as is. Simplifying, when possible, not only makes the expression neater but also easier to interpret or use in further calculations.