Sonji Correctly Added $\frac{2p+1}{p-8}$ And $\frac{3p-3}{4p}$ And Got $\frac{8p^2+4p+3p^2-27p+24}{(p-8)(4p)}$. What Is The Simplified Sum?A. \$\frac{5p^2-23p+24}{(p-8)(4p)}$[/tex\]B.

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying complex expressions. In this article, we will guide you through the process of simplifying the given expression $\frac{2p+1}{p-8} + \frac{3p-3}{4p}$.

Understanding the Problem

The problem requires us to simplify the given expression by adding the two fractions. To do this, we need to find a common denominator and then combine the fractions.

Finding a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the denominators. In this case, the denominators are p−8p-8 and 4p4p. The LCM of these two expressions is (p−8)(4p)(p-8)(4p).

Simplifying the Expression

Now that we have found the common denominator, we can rewrite the expression as:

2p+1p−8+3p−34p=(2p+1)(4p)+(3p−3)(p−8)(p−8)(4p)\frac{2p+1}{p-8} + \frac{3p-3}{4p} = \frac{(2p+1)(4p) + (3p-3)(p-8)}{(p-8)(4p)}

Expanding the Numerator

To simplify the expression further, we need to expand the numerator. We can do this by multiplying the terms in the numerator.

(2p+1)(4p)=8p2+4p(2p+1)(4p) = 8p^2 + 4p

(3p−3)(p−8)=3p2−24p−3p+24=3p2−27p+24(3p-3)(p-8) = 3p^2 - 24p - 3p + 24 = 3p^2 - 27p + 24

Combining the Terms

Now that we have expanded the numerator, we can combine the terms.

8p2+4p+3p2−27p+24=11p2−23p+248p^2 + 4p + 3p^2 - 27p + 24 = 11p^2 - 23p + 24

Simplifying the Expression

Now that we have combined the terms, we can simplify the expression by writing it in the form of a single fraction.

8p2+4p+3p2−27p+24(p−8)(4p)=11p2−23p+24(p−8)(4p)\frac{8p^2 + 4p + 3p^2 - 27p + 24}{(p-8)(4p)} = \frac{11p^2 - 23p + 24}{(p-8)(4p)}

Conclusion

In conclusion, the simplified sum of the given expression is $\frac{11p^2 - 23p + 24}{(p-8)(4p)}$. This is the final answer to the problem.

Final Answer

The final answer is: 11p2−23p+24(p−8)(4p)\boxed{\frac{11p^2 - 23p + 24}{(p-8)(4p)}}

Discussion

This problem requires a good understanding of algebraic expressions and the ability to simplify complex expressions. The key steps involved in simplifying the expression are finding a common denominator, expanding the numerator, and combining the terms.

Tips and Tricks

  • When simplifying algebraic expressions, it's essential to identify the common denominator and rewrite the expression accordingly.
  • Expanding the numerator can be a challenging task, but it's crucial to get it right to simplify the expression.
  • Combining the terms is the final step in simplifying the expression, and it requires careful attention to detail.

Related Problems

  • Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to practice regularly to become proficient.
  • The key to simplifying algebraic expressions is to identify the common denominator and rewrite the expression accordingly.
  • Expanding the numerator and combining the terms are the final steps in simplifying the expression.

Common Mistakes

  • One common mistake when simplifying algebraic expressions is to forget to identify the common denominator.
  • Another common mistake is to expand the numerator incorrectly, which can lead to incorrect results.
  • Finally, combining the terms incorrectly can also lead to incorrect results.

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying complex expressions. By following the steps outlined in this article, you can simplify algebraic expressions with ease. Remember to identify the common denominator, expand the numerator, and combine the terms carefully to get the correct result.

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying complex expressions. In this article, we will provide a Q&A guide to help you understand the process of simplifying algebraic expressions.

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to identify the common denominator. This is the least common multiple (LCM) of the denominators in the expression.

Q: How do I find the common denominator?

A: To find the common denominator, you need to identify the least common multiple (LCM) of the denominators in the expression. You can do this by listing the multiples of each denominator and finding the smallest multiple that is common to both.

Q: What is the next step in simplifying an algebraic expression?

A: The next step in simplifying an algebraic expression is to expand the numerator. This involves multiplying the terms in the numerator to get a single expression.

Q: How do I expand the numerator?

A: To expand the numerator, you need to multiply the terms in the numerator. This involves using the distributive property to multiply each term in the numerator by each term in the other expression.

Q: What is the final step in simplifying an algebraic expression?

A: The final step in simplifying an algebraic expression is to combine the terms. This involves adding or subtracting the terms in the numerator to get a single expression.

Q: How do I combine the terms?

A: To combine the terms, you need to add or subtract the terms in the numerator. This involves using the rules of arithmetic to combine the terms.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include forgetting to identify the common denominator, expanding the numerator incorrectly, and combining the terms incorrectly.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including solving equations, graphing functions, and modeling real-world phenomena.

Q: How can I use simplifying algebraic expressions in my everyday life?

A: You can use simplifying algebraic expressions in your everyday life by applying mathematical concepts to real-world problems. For example, you can use algebraic expressions to model the cost of goods, the time it takes to complete a task, or the distance between two points.

Q: What are some tips for simplifying algebraic expressions?

A: Some tips for simplifying algebraic expressions include:

  • Identifying the common denominator before expanding the numerator
  • Using the distributive property to expand the numerator
  • Combining the terms carefully to avoid errors
  • Checking your work to ensure that the expression is simplified correctly

Q: How can I use technology to simplify algebraic expressions?

A: You can use technology, such as calculators or computer software, to simplify algebraic expressions. This can be especially helpful when working with complex expressions or when you need to check your work.

Q: What are some common algebraic expressions that require simplification?

A: Some common algebraic expressions that require simplification include:

  • Fractions with unlike denominators
  • Expressions with multiple terms in the numerator or denominator
  • Expressions with variables in the numerator or denominator
  • Expressions with exponents or roots

Q: How can I simplify expressions with exponents or roots?

A: To simplify expressions with exponents or roots, you need to use the rules of exponents and roots. This involves using the product rule, quotient rule, and power rule to simplify the expression.

Q: What are some tips for simplifying expressions with exponents or roots?

A: Some tips for simplifying expressions with exponents or roots include:

  • Using the product rule to simplify expressions with multiple terms
  • Using the quotient rule to simplify expressions with fractions
  • Using the power rule to simplify expressions with exponents
  • Checking your work to ensure that the expression is simplified correctly

Conclusion

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the steps involved in simplifying complex expressions. By following the steps outlined in this article, you can simplify algebraic expressions with ease. Remember to identify the common denominator, expand the numerator, and combine the terms carefully to get the correct result.