Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 8: Problem 195
In the following exercises, add. $$ \frac{8}{t+5}+\frac{6}{t-5} $$
Short Answer
Expert verified
\(\frac{14t - 10}{(t+5)(t-5)}\)
Step by step solution
01
- Identify the Common Denominator
To add two fractions, they must have a common denominator. The denominators of the fractions \(\frac{8}{t+5}\) and \(\frac{6}{t-5}\) are \(t+5\) and \(t-5\), respectively. The common denominator will be the product of these two terms: \((t+5)(t-5)\).
02
- Rewrite Each Fraction with the Common Denominator
Rewrite each fraction so that they both have the common denominator \((t+5)(t-5)\). For the first fraction, multiply both the numerator and the denominator by \(t-5\):\[\frac{8(t-5)}{(t+5)(t-5)}\]For the second fraction, multiply both the numerator and the denominator by \(t+5\):\[\frac{6(t+5)}{(t+5)(t-5)}\]
03
- Add the Fractions
Now that both fractions have the same denominator \((t+5)(t-5)\), add the numerators together while keeping the common denominator:\[\frac{8(t-5) + 6(t+5)}{(t+5)(t-5)}\]
04
- Simplify the Numerator
Expand and simplify the numerator:\[8(t-5) + 6(t+5) = 8t - 40 + 6t + 30 = 14t - 10\]The resulting fraction is:\[\frac{14t - 10}{(t+5)(t-5)}\]
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
common denominator
When adding fractions, it鈥檚 important to have a common denominator. The common denominator is the shared multiple of the denominators of the fractions. In this exercise, the fractions are \(\frac{8}{t+5}\) and \(\frac{6}{t-5}\). Their denominators are \((t+5)\) and \((t-5)\), which are different. To find the common denominator, follow these steps:
- 1. List the denominators: \((t+5)\) and \((t-5)\)
- 2. Multiply the denominators together: \((t+5)(t-5)\)
rational expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. In our exercise, \(\frac{8}{t+5}\) and \(\frac{6}{t-5}\) are both rational expressions because both the numerators (8 and 6) and the denominators (\(t+5\) and \(t-5\)) are polynomials. Understanding rational expressions involves recognizing:
- The relationship between the numerator and the denominator
- How to manipulate these polynomials to perform operations such as addition, subtraction, multiplication, and division
- The importance of domain restrictions, i.e., the values that the variable can take to avoid division by zero
fraction addition
Adding fractions involves the following steps: finding a common denominator, rewriting the fractions with that common denominator, adding the numerators, and simplifying the result. Here's how it looks in our exercise:
- 1. Find a common denominator: \((t+5)(t-5)\)
- 2. Rewrite each fraction:
For \(\frac{8}{t+5}\), multiply both the numerator and the denominator by \(t-5\): \(\frac{8(t-5)}{(t+5)(t-5)}\)
For \(\frac{6}{t-5}\), multiply both the numerator and the denominator by \(t+5\): \(\frac{6(t+5)}{(t+5)(t-5)}\) - 3. Add the rewritten fractions: \(\frac{8(t-5) + 6(t+5)}{(t+5)(t-5)}\)
- 4. Combine and simplify the numerators: \(\frac{8t - 40 + 6t + 30}{(t+5)(t-5)} = \frac{14t - 10}{(t+5)(t-5)}\)
simplifying expressions
Simplifying expressions is an essential part of working with rational expressions. Once we reach a fraction where the numerator and denominator have common factors, we can potentially simplify further. In our final step:
- Expand the simplified numerator: \(8t - 40 + 6t + 30 = 14t - 10\)
- Write the expanded numerator over the common denominator: \( \frac{14t - 10}{(t+5)(t-5)}\)
