Add or subtract, as indicated. See Example 4. (1/6m) + (2/5m) + (4/m)

Verified step by step guidance

1. Identify the terms to be added: \(\frac{1}{6m}\), \(\frac{2}{5m}\), and \(\frac{4}{m}\). Since all terms have the variable \(m\) in the denominator, we will treat \(m\) as a common factor in the denominators.
2. Find the least common denominator (LCD) for the fractions. The denominators are \$6m\(, \)5m\(, and \)m\(. The LCD for the numeric parts 6, 5, and 1 is the least common multiple (LCM) of 6 and 5, which is 30. Since all denominators have \)m\(, the LCD is \)30m$.
3. Rewrite each fraction with the common denominator \$30m$ by multiplying numerator and denominator appropriately: - For \(\frac{1}{6m}\), multiply numerator and denominator by 5 to get \(\frac{5}{30m}\). - For \(\frac{2}{5m}\), multiply numerator and denominator by 6 to get \(\frac{12}{30m}\). - For \(\frac{4}{m}\), multiply numerator and denominator by 30 to get \(\frac{120}{30m}\).
4. Add the numerators over the common denominator: \(\frac{5}{30m} + \frac{12}{30m} + \frac{120}{30m} = \frac{5 + 12 + 120}{30m}\).
5. Combine the numerators to get a single fraction: \(\frac{137}{30m}\). This is the simplified sum of the original expression.

Key Concepts

Adding and Subtracting Algebraic Fractions

To add or subtract algebraic fractions, first find a common denominator that includes all variable and numerical factors. Then, rewrite each fraction with this common denominator before combining the numerators. Simplify the resulting expression if possible.

Finding the Least Common Denominator (LCD)

The least common denominator is the smallest expression that all denominators divide into evenly. For algebraic fractions, this involves factoring numerical coefficients and variables, then taking the highest powers of each factor to form the LCD.

Simplifying Algebraic Expressions

After combining fractions, simplify the expression by factoring and reducing common terms in the numerator and denominator. This step ensures the final answer is in its simplest form, making it easier to interpret and use.