Problem 63 - Algebra 2

Question

Simplify each expression. \(\frac{-2}{a b}+\frac{5}{a^{2}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-2a + 5b}{a^2b}\).

Step 1: Identify the Common Denominator

To add the two fractions, we need a common denominator. The denominators are \(ab\) and \(a^2\). The least common denominator (LCD) is \(a^2b\) because it can be divided by both original denominators.

Step 2: Rewrite the Fractions with the Common Denominator

Convert each fraction to have the \(a^2b\) denominator. Rewrite the first fraction by multiplying both the numerator and the denominator by \(a\):\[\frac{-2}{ab} = \frac{-2 \cdot a}{ab \cdot a} = \frac{-2a}{a^2b}\]Rewrite the second fraction by multiplying both the numerator and the denominator by \(b\):\[\frac{5}{a^2} = \frac{5 \cdot b}{a^2 \cdot b} = \frac{5b}{a^2b}\]

Step 3: Add the Fractions Together

Now that both fractions have the common denominator \(a^2b\), add them by combining their numerators:\[\frac{-2a}{a^2b} + \frac{5b}{a^2b} = \frac{-2a + 5b}{a^2b}\]

Step 4: Simplify if Possible

Check the final expression \(\frac{-2a + 5b}{a^2b}\) to see if it can be simplified further. No common factors exist in the numerator and denominator, so this is the simplified expression.

Key Concepts

Least Common Denominator

When dealing with the addition of fractions, finding the least common denominator (LCD) is an essential step. The LCD is the smallest number or expression that both denominators can divide into evenly. It ensures that you can add fractions correctly by having the same denominator.
In our exercise, the denominators are different, namely \(ab\) and \(a^2\). To find the LCD, we need to determine the lowest power of each variable and factor that captures both denominators.
Here, the least common denominator is \(a^2b\), as it is the smallest expression that \(ab\) and \(a^2\) can both divide into without a remainder. Understanding how to find the LCD is crucial for simplifying expressions correctly.

Fractions

Fractions represent parts of a whole and consist of a numerator and a denominator. In mathematics, fractions are used to express division of numbers or variables.
For example, \(\frac{-2}{ab}\) tells us that \(-2\) is to be divided by \(ab\), while \(\frac{5}{a^2}\) expresses division of \(5\) by \(a^2\).
Dealing with fractions often involves performing operations like addition, subtraction, multiplication, and division.

• The numerator is the top part of the fraction, which tells you how many parts we have.
• The denominator is the bottom part, which tells you how many parts the whole is divided into.

Mastering fractions is key to solving algebraic expressions successfully.

Addition of Fractions

Adding fractions involves making sure that both fractions have the same denominator. Once you have the least common denominator (LCD), you can convert each fraction to an equivalent fraction with this common denominator.
In the original exercise, we had the fractions \(\frac{-2}{ab}\) and \(\frac{5}{a^2}\). By rewriting them with the common denominator \(a^2b\), we get \(\frac{-2a}{a^2b}\) and \(\frac{5b}{a^2b}\).
Now, adding fractions becomes simple: you only need to add their numerators, while keeping their common denominator unchanged.

• Add numerators: -2a and 5b result in -2a + 5b.
• Keep the denominator as \(a^2b\).

Thus, the result is \(\frac{-2a + 5b}{a^2b}\). Remember, having a common denominator is necessary for adding any fractions.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form, where no further simplification can be done. In terms of fractions, it involves checking if the numerator and denominator have any common factors that can be divided out.
For the expression \(\frac{-2a + 5b}{a^2b}\), inspect the numerator and denominator.
Check if there are common factors that can be simplified, although in this case, there are none.

• The numerator (-2a + 5b) shares no common factors with the denominator (a^2b).

Thus, \(\frac{-2a + 5b}{a^2b}\) is already presented in its simplest form. Understanding how to simplify helps in keeping expressions concise and as easy as possible to interpret or use in further computations.