Problem 26 - Algebra 1: Concepts and Skills

Question

Write the sum in simplest form. $$ \frac{3}{12 m^{3}}+\frac{m+1}{4 m^{3}} $$

Step-by-Step Solution

Answer

The simplified form of the given sum is \(\frac{m+2}{4m^{3}}\).

Step 1: Simplify the First Fraction

The first fraction \(\frac{3}{12m^{3}}\) can be simplified by dividing the numerator and the denominator by 3 to get \(\frac{1}{4m^{3}}\).

Step 2: Add the Fractions

Since the denominators of both fractions are the same (\(4m^{3}\)), they can be added together as: \(\frac{1}{4m^{3}}+\frac{m+1}{4m^{3}}\) to get \(\frac{1+m+1}{4m^{3}}\).

Step 3: Simplify further

Combine like terms in the numerator, so the sum becomes \(\frac{m+2}{4m^{3}}\).

Key Concepts

Simplification

When we first look at an algebraic fraction, it might seem a bit intimidating, especially if it involves numerous variables and coefficients. This is where simplification steps in to make things easier.
Simplification in mathematics is all about making expressions less complex, and this often involves reducing fractions to their simplest form. To simplify a fraction, we examine both the numerator and the denominator to see if they share a common factor – a number or variable both parts can be divided by without leaving remainders. In many cases, as seen in our exercise, the fraction \( \frac{3}{12m^{3}} \) needs simplification. Each term of the numerator and the denominator in this fraction was carefully divided by their greatest common factor, which is 3. This reduces the fraction to \( \frac{1}{4m^{3}} \).
This streamlined version is much easier to work with in subsequent mathematical operations.

Addition of Fractions

Adding fractions in algebra follows the same basic rules as adding numerical fractions. The key is ensuring that the fractions have the same denominators.
Let's take a closer look at the fractions from the exercise: \( \frac{1}{4m^{3}} \) and \( \frac{m+1}{4m^{3}} \). Both align with the same denominator, \( 4m^{3} \). This means we're good to go for direct addition.When the denominators are identical, you maintain the denominator and simply add the numerators together. So, \( \frac{1}{4m^{3}} + \frac{m+1}{4m^{3}} \) transforms into \( \frac{1 + (m+1)}{4m^{3}} \).
This step is straightforward yet vital because adding the numerators while keeping the denominators unchanged preserves the balance of the fraction.

Common Denominator

To successfully add algebraic fractions, a common denominator is essential. The common denominator provides a shared base for the numerators over which they can seamlessly combine.Fractions \( \frac{1}{4m^{3}} \) and \( \frac{m+1}{4m^{3}} \) from the exercise already share this common denominator, making the addition process more direct and efficient. Without a common denominator, we'd need to find it by multiplying the denominators until both fractions match, which can be a bit of extra work.
However, when fractions have the same denominator from the get-go, like in our scenario, the process is simplified. We directly add the numerators because they fit snuggly over that single denominator. This approach ensures the expression remains balanced and accurate.
Always double-check your denominators before adding fractions, as this can save you time and reduce errors.