Question

Express \(\frac{1}{s}+\frac{1}{s^{2}}\) as a single fraction.

Short answer

Question: Combine the two fractions \(\frac{1}{s}\) and \(\frac{1}{s^{2}}\) into a single fraction. Answer: \(\frac{s+1}{s^2}\)

Step-by-step solution

Find the common denominator

The common denominator must be a multiple of both denominators, \(s\) and \(s^2\). Since \(s^2\) is a multiple of \(s\), the least common denominator (LCD) is \(s^2\).

Express both fractions with the common denominator

Now, we need to rewrite each fraction with the LCD (\(s^2\)) as the denominator. For the first fraction, we multiply both the numerator and denominator by \(s\) to get \(\frac{1}{s}\cdot\frac{s}{s}=\frac{s}{s^2}\). The second fraction already has the common denominator, so it remains the same: \(\frac{1}{s^2}\)

Add the fractions

Since the fractions now have a common denominator of \(s^2\), we can add the numerators together and keep the denominator the same. So, \(\frac{s}{s^2}+\frac{1}{s^2}=\frac{s+1}{s^2}\) The combined single fraction for the given expression is \(\frac{s+1}{s^2}\).

Key concepts

Understanding the Common Denominator

When dealing with algebraic fractions, finding a common denominator is crucial. A common denominator allows us to combine fractions easily, a lot like what you might already be familiar with in regular fractions. Here, our task is to ensure all fractions involved share a matching denominator that allows for straightforward addition.

For example, in the expression \(\frac{1}{s}+\frac{1}{s^2}\), we need to determine a common denominator. Since \(s^2\) is a multiple of \(s\), \(s^2\) becomes our least common denominator (LCD). This step of finding the LCD involves identifying the smallest expression that both individual denominators divide into perfectly. This little adjustment paves the way for seamless fraction operations, ensuring no terms are left behind.

Adding Fractions with a Common Denominator

Once we've harmonized the denominators, fraction addition becomes a simple endeavor. Using a common denominator allows us to add the numerators directly instead of dealing with complicated calculations.

In our example, after adjusting \(\frac{1}{s}\) to \(\frac{s}{s^2}\), both fractions become \(\frac{s}{s^2}\) and \(\frac{1}{s^2}\). They share the denominator \(s^2\), ready for straightforward addition.

We then simply combine the numerators: the expression turns into \(\frac{s+1}{s^2}\). This method emphasizes the importance of aligning denominators to make further operations direct and less cumbersome. It’s almost like putting all your groceries into one bag, making it easier to carry.

Simplifying Expressions After Addition

After adding fractions and landing on a result like \(\frac{s+1}{s^2}\), the next step is to check if the expression can be simplified further. Simplifying expressions can mean several things: reducing fractions, canceling common factors, and more.

In this specific case, \(\frac{s+1}{s^2}\) is already simplified. There's no common factor between the numerator \(s+1\) and the denominator \(s^2\) other than 1. Simplification would involve dividing the numerator and the denominator by the greatest common divisor if one existed.

Maintaining a clean, reduced expression is important for clarity and accuracy in algebra. It ensures your answer is as neat and usable as possible for any further calculations you might need to do.