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Chapter 7: Problem 5

Perform the indicated operation. Simplify, if possible. \(\frac{4}{x}+\frac{9}{x^{2}}\)

Short Answer

Expert verified
\(\frac{10}{x}\)

Step by step solution

01

Identify Common Denominators

Observe that the denominators in each fraction are the same. Both fractions have a denominator of \(x\).
02

Add the Numerators

Since the denominators are the same, we can add the numerators directly: \(6 + 4\).
03

Simplify the Sum

Add the numerators to get: \(6 + 4 = 10\).
04

Write the Result

Combine the simplified numerator with the common denominator to write the final fraction: \(\frac{10}{x}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Denominators
When adding or subtracting fractions, the first step is to find a common denominator. The common denominator is the same bottom number in both fractions. This makes it possible to combine the fractions.
In our example, both fractions already have the same denominator, which is `x`. There's no need to change anything because they are ready to be added!
Adding Fractions
Once you have common denominators, the next step is to add the fractions. Focus only on the numerators, which are the top parts of the fractions.
In our exercise, we add the numerators `6` and `4`. Since they share the same denominator `x`, you just add them directly: \(6 + 4\), which equals `10`.
After adding, you keep the denominator the same, which means you write the result as \(\frac{10}{x}\).
This method keeps everything in order and ensures your fractions are combined correctly.
Simplifying Fractions
After adding, it鈥檚 always a good idea to simplify the result if possible. Simplifying means making the fraction as simple as it can be.
In our case, \(\frac{10}{x}\) can't be simplified further unless `x` and `10` share a common factor. If, for example, `x` was `5`, we could simplify the fraction to \(\frac{10}{5} = 2\).
Always check if the numerator and denominator have common factors, and if they do, divide them by the greatest common divisor to simplify!
Algebraic Expressions
Fractions often contain algebraic expressions, like in our exercise where `x` is a variable. Working with variables might seem tricky, but the same rules of fraction operations apply!
When dealing with algebraic fractions, always:
  • Ensure you have a common denominator if adding or subtracting
  • Combine numerators while keeping the denominator the same
  • Simplify the fraction if possible by factoring the numerator and denominator
Understanding these steps makes handling algebraic fractions straightforward and helps you solve a wide range of math problems.
Keep practicing and these steps will become second nature!

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