Add And Subtract, Then Simplify Completely: $\[ \frac{p+7}{p-6} + \frac{p-1}{p-9} \\]Enter The Numerator And Denominator Separately In The Boxes Below. If The Denominator Is 1, Enter The Number 1. Do Not Leave Either Box Blank.Answer:-

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Understanding Complex Fractions

Complex fractions are mathematical expressions that contain fractions within other fractions. They can be challenging to simplify, but with a clear understanding of the steps involved, you can master the process. In this article, we will focus on simplifying the complex fraction p+7p−6+p−1p−9\frac{p+7}{p-6} + \frac{p-1}{p-9}.

The Importance of Simplifying Complex Fractions

Simplifying complex fractions is crucial in mathematics, as it helps to:

  • Reduce errors: Simplifying complex fractions reduces the likelihood of errors, making it easier to work with mathematical expressions.
  • Improve understanding: Simplifying complex fractions helps to clarify the underlying mathematical concepts, making it easier to understand and apply them.
  • Enhance problem-solving skills: Simplifying complex fractions requires a deep understanding of mathematical concepts and techniques, which can help to improve problem-solving skills.

Step 1: Find a Common Denominator

To simplify the complex fraction p+7p−6+p−1p−9\frac{p+7}{p-6} + \frac{p-1}{p-9}, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators p−6p-6 and p−9p-9. To find the LCM, we can list the multiples of each denominator:

  • Multiples of p−6p-6: (p−6),2(p−6),3(p−6),...(p-6), 2(p-6), 3(p-6), ...
  • Multiples of p−9p-9: (p−9),2(p−9),3(p−9),...(p-9), 2(p-9), 3(p-9), ...

The first multiple that appears in both lists is the LCM. In this case, the LCM is (p−6)(p−9)(p-6)(p-9).

Step 2: Rewrite the Complex Fraction with the Common Denominator

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

p+7p−6+p−1p−9=(p+7)(p−9)+(p−1)(p−6)(p−6)(p−9)\frac{p+7}{p-6} + \frac{p-1}{p-9} = \frac{(p+7)(p-9) + (p-1)(p-6)}{(p-6)(p-9)}

Step 3: Simplify the Numerator

To simplify the numerator, we can expand the expressions (p+7)(p−9)(p+7)(p-9) and (p−1)(p−6)(p-1)(p-6):

  • (p+7)(p−9)=p2−9p+7p−63=p2−2p−63(p+7)(p-9) = p^2 - 9p + 7p - 63 = p^2 - 2p - 63
  • (p−1)(p−6)=p2−6p−p+6=p2−7p+6(p-1)(p-6) = p^2 - 6p - p + 6 = p^2 - 7p + 6

Now, we can substitute these expressions into the numerator:

(p2−2p−63)+(p2−7p+6)(p−6)(p−9)\frac{(p^2 - 2p - 63) + (p^2 - 7p + 6)}{(p-6)(p-9)}

Step 4: Combine Like Terms

To simplify the numerator further, we can combine like terms:

  • p2−2p−63+p2−7p+6=2p2−9p−57p^2 - 2p - 63 + p^2 - 7p + 6 = 2p^2 - 9p - 57

Now, we can substitute this expression into the numerator:

2p2−9p−57(p−6)(p−9)\frac{2p^2 - 9p - 57}{(p-6)(p-9)}

Step 5: Factor the Numerator

To simplify the numerator further, we can factor the expression 2p2−9p−572p^2 - 9p - 57:

  • 2p2−9p−57=(2p+3)(p−19)2p^2 - 9p - 57 = (2p + 3)(p - 19)

Now, we can substitute this expression into the numerator:

(2p+3)(p−19)(p−6)(p−9)\frac{(2p + 3)(p - 19)}{(p-6)(p-9)}

Step 6: Simplify the Complex Fraction

Now that we have simplified the numerator, we can simplify the complex fraction:

(2p+3)(p−19)(p−6)(p−9)=2p2−35p−57(p−6)(p−9)\frac{(2p + 3)(p - 19)}{(p-6)(p-9)} = \frac{2p^2 - 35p - 57}{(p-6)(p-9)}

Conclusion

Simplifying complex fractions requires a deep understanding of mathematical concepts and techniques. By following the steps outlined in this article, you can simplify complex fractions and improve your problem-solving skills. Remember to find a common denominator, rewrite the complex fraction with the common denominator, simplify the numerator, combine like terms, factor the numerator, and simplify the complex fraction.

Final Answer

The final answer is:

  • Numerator: 2p2−35p−572p^2 - 35p - 57
  • Denominator: (p−6)(p−9)(p-6)(p-9)

Understanding Complex Fractions

Complex fractions are mathematical expressions that contain fractions within other fractions. They can be challenging to simplify, but with a clear understanding of the steps involved, you can master the process. In this article, we will focus on simplifying the complex fraction p+7p−6+p−1p−9\frac{p+7}{p-6} + \frac{p-1}{p-9}.

Q&A: Simplifying Complex Fractions

Q: What is a complex fraction?

A: A complex fraction is a mathematical expression that contains fractions within other fractions.

Q: Why is it important to simplify complex fractions?

A: Simplifying complex fractions is crucial in mathematics, as it helps to reduce errors, improve understanding, and enhance problem-solving skills.

Q: How do I find a common denominator for complex fractions?

A: To find a common denominator, list the multiples of each denominator and find the least common multiple (LCM).

Q: What is the least common multiple (LCM)?

A: The LCM is the smallest multiple that appears in both lists of multiples.

Q: How do I rewrite a complex fraction with a common denominator?

A: Rewrite the complex fraction by multiplying the numerator and denominator by the common denominator.

Q: What is the next step after finding a common denominator?

A: The next step is to simplify the numerator by combining like terms and factoring.

Q: How do I simplify the numerator?

A: Simplify the numerator by combining like terms and factoring.

Q: What is the final step in simplifying a complex fraction?

A: The final step is to simplify the complex fraction by dividing the numerator and denominator by their greatest common divisor (GCD).

Q: What is the greatest common divisor (GCD)?

A: The GCD is the largest number that divides both the numerator and denominator.

Q: Why is it important to simplify complex fractions?

A: Simplifying complex fractions is crucial in mathematics, as it helps to reduce errors, improve understanding, and enhance problem-solving skills.

Q: Can you provide an example of simplifying a complex fraction?

A: Yes, the example provided earlier is a complex fraction that can be simplified using the steps outlined in this article.

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

  • Not finding a common denominator
  • Not simplifying the numerator
  • Not factoring the numerator
  • Not simplifying the complex fraction

Q: How can I practice simplifying complex fractions?

A: Practice simplifying complex fractions by working through examples and exercises.

Conclusion

Simplifying complex fractions requires a deep understanding of mathematical concepts and techniques. By following the steps outlined in this article and practicing simplifying complex fractions, you can improve your problem-solving skills and become more confident in your ability to simplify complex fractions.

Final Answer

The final answer is:

  • Numerator: 2p2−35p−572p^2 - 35p - 57
  • Denominator: (p−6)(p−9)(p-6)(p-9)