Perform The Indicated Operation: 11 D − 4 + 4 D + 1 \frac{11}{d-4}+\frac{4}{d+1} D − 4 11 ​ + D + 1 4 ​

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Introduction

When working with algebraic expressions, it's not uncommon to encounter fractions that need to be combined. In this article, we'll focus on simplifying the expression 11d−4+4d+1\frac{11}{d-4}+\frac{4}{d+1}, which involves adding two fractions with different denominators. By following a step-by-step approach, we'll break down the process of combining fractions and provide a clear understanding of the underlying concepts.

Understanding the Problem

The given expression is a sum of two fractions: 11d−4\frac{11}{d-4} and 4d+1\frac{4}{d+1}. To simplify this expression, we need to find a common denominator, which will allow us to add the fractions together. The common denominator is the least common multiple (LCM) of the two denominators, d−4d-4 and d+1d+1.

Finding the Least Common Multiple (LCM)

To find the LCM of d−4d-4 and d+1d+1, we need to consider the factors of each expression. The factors of d−4d-4 are 11, d−4d-4, and (d−4)(d+1)(d-4)(d+1). The factors of d+1d+1 are 11, d+1d+1, and (d−4)(d+1)(d-4)(d+1). Since both expressions share the factor (d−4)(d+1)(d-4)(d+1), we can conclude that the LCM is (d−4)(d+1)(d-4)(d+1).

Rewriting the Fractions with the Common Denominator

Now that we have the common denominator, we can rewrite each fraction with the new denominator. To do this, we need to multiply the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.

For the first fraction, 11d−4\frac{11}{d-4}, we need to multiply the numerator and denominator by d+1d+1 to obtain:

11(d+1)(d−4)(d+1)\frac{11(d+1)}{(d-4)(d+1)}

For the second fraction, 4d+1\frac{4}{d+1}, we need to multiply the numerator and denominator by d−4d-4 to obtain:

4(d−4)(d−4)(d+1)\frac{4(d-4)}{(d-4)(d+1)}

Combining the Fractions

Now that we have rewritten each fraction with the common denominator, we can combine them by adding the numerators:

11(d+1)(d−4)(d+1)+4(d−4)(d−4)(d+1)=11(d+1)+4(d−4)(d−4)(d+1)\frac{11(d+1)}{(d-4)(d+1)} + \frac{4(d-4)}{(d-4)(d+1)} = \frac{11(d+1) + 4(d-4)}{(d-4)(d+1)}

Simplifying the Expression

To simplify the expression, we can expand the numerator and combine like terms:

11(d+1)+4(d−4)(d−4)(d+1)=11d+11+4d−16(d−4)(d+1)\frac{11(d+1) + 4(d-4)}{(d-4)(d+1)} = \frac{11d + 11 + 4d - 16}{(d-4)(d+1)}

=15d−5(d−4)(d+1)= \frac{15d - 5}{(d-4)(d+1)}

Conclusion

In this article, we simplified the expression 11d−4+4d+1\frac{11}{d-4}+\frac{4}{d+1} by finding the least common multiple (LCM) of the two denominators and rewriting each fraction with the common denominator. We then combined the fractions by adding the numerators and simplified the resulting expression. By following this step-by-step approach, we can confidently simplify algebraic expressions involving fractions.

Real-World Applications

Simplifying algebraic expressions involving fractions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, the expression 11d−4+4d+1\frac{11}{d-4}+\frac{4}{d+1} may represent the motion of an object with two different velocities. By simplifying the expression, we can gain a deeper understanding of the object's motion and make more accurate predictions.

Tips and Tricks

When simplifying algebraic expressions involving fractions, it's essential to follow these tips and tricks:

  • Find the least common multiple (LCM): The LCM is the key to simplifying expressions involving fractions. Make sure to find the LCM of the two denominators before rewriting the fractions.
  • Rewrite the fractions with the common denominator: Once you have the LCM, rewrite each fraction with the new denominator. This will allow you to combine the fractions by adding the numerators.
  • Combine like terms: When simplifying the expression, make sure to combine like terms in the numerator. This will help you arrive at the final simplified expression.

By following these tips and tricks, you'll be well on your way to simplifying algebraic expressions involving fractions like a pro!

Introduction

In our previous article, we explored the process of simplifying algebraic expressions involving fractions. We walked through a step-by-step approach to combine fractions and arrived at a simplified expression. In this article, we'll answer some frequently asked questions (FAQs) related to simplifying algebraic expressions involving fractions.

Q&A

Q: What is the least common multiple (LCM) and why is it important?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of simplifying algebraic expressions involving fractions, the LCM is essential because it allows us to rewrite each fraction with a common denominator. This, in turn, enables us to combine the fractions by adding the numerators.

Q: How do I find the LCM of two expressions?

A: To find the LCM of two expressions, you can list the factors of each expression and identify the common factors. The LCM is then the product of the highest power of each common factor. For example, if we have the expressions d−4d-4 and d+1d+1, we can list their factors as follows:

  • d−4d-4: 11, d−4d-4, (d−4)(d+1)(d-4)(d+1)
  • d+1d+1: 11, d+1d+1, (d−4)(d+1)(d-4)(d+1)

The LCM is then (d−4)(d+1)(d-4)(d+1).

Q: What if the LCM is not a simple expression?

A: If the LCM is not a simple expression, you may need to use algebraic manipulation to simplify it. For example, if the LCM is (d−4)(d+1)(d-4)(d+1), you can expand it as follows:

(d−4)(d+1)=d2−4d+d−4=d2−3d−4(d-4)(d+1) = d^2 - 4d + d - 4 = d^2 - 3d - 4

Q: Can I simplify an expression with multiple fractions?

A: Yes, you can simplify an expression with multiple fractions by following the same steps as before. First, find the LCM of the denominators, then rewrite each fraction with the common denominator, and finally combine the fractions by adding the numerators.

Q: What if I have a fraction with a variable in the denominator?

A: If you have a fraction with a variable in the denominator, you can simplify it by following the same steps as before. However, you may need to use algebraic manipulation to simplify the expression. For example, if you have the fraction 1d2−4\frac{1}{d^2 - 4}, you can simplify it as follows:

1d2−4=1(d−2)(d+2)\frac{1}{d^2 - 4} = \frac{1}{(d-2)(d+2)}

Q: Can I use a calculator to simplify an expression?

A: Yes, you can use a calculator to simplify an expression. However, it's essential to understand the underlying math concepts and be able to simplify expressions manually. This will help you develop problem-solving skills and ensure that you understand the math behind the calculations.

Conclusion

Simplifying algebraic expressions involving fractions is a crucial skill in mathematics and has numerous real-world applications. By following the steps outlined in this article and answering the FAQs, you'll be well on your way to simplifying expressions like a pro! Remember to find the least common multiple (LCM), rewrite each fraction with the common denominator, and combine the fractions by adding the numerators. With practice and patience, you'll become proficient in simplifying algebraic expressions involving fractions.

Tips and Tricks

  • Practice, practice, practice: The more you practice simplifying algebraic expressions involving fractions, the more comfortable you'll become with the process.
  • Use algebraic manipulation: Algebraic manipulation is a powerful tool for simplifying expressions. Don't be afraid to use it to simplify complex expressions.
  • Check your work: Always check your work to ensure that you've simplified the expression correctly.
  • Use a calculator: If you're struggling to simplify an expression manually, consider using a calculator to check your work.

By following these tips and tricks, you'll be able to simplify algebraic expressions involving fractions with ease!