Simplify The Expression:$\[ \frac{(x-395)^2}{100}-\frac{x-5}{(x-500)^2} \\]

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Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in calculus, algebra, and other advanced mathematical disciplines. These expressions often involve multiple variables, fractions, and exponents, making them challenging to simplify. In this article, we will focus on simplifying the given expression: (x−395)2100−x−5(x−500)2\frac{(x-395)^2}{100}-\frac{x-5}{(x-500)^2}. We will break down the process into manageable steps, using various mathematical techniques to simplify the expression.

Understanding the Expression

The given expression is a combination of two fractions, each with a different denominator. The first fraction has a quadratic expression in the numerator, while the second fraction has a linear expression in the numerator. To simplify this expression, we need to find a common denominator and combine the fractions.

Step 1: Find a Common Denominator

To find a common denominator, we need to identify the least common multiple (LCM) of the two denominators. In this case, the denominators are 100100 and (x−500)2(x-500)^2. The LCM of these two expressions is 100(x−500)2100(x-500)^2.

Step 2: Rewrite the Fractions with the Common Denominator

Now that we have found the common denominator, we can rewrite each fraction with the common denominator.

(x−395)2100=(x−395)2⋅(x−500)2100(x−500)2\frac{(x-395)^2}{100} = \frac{(x-395)^2 \cdot (x-500)^2}{100(x-500)^2}

x−5(x−500)2=(x−5)⋅100100(x−500)2\frac{x-5}{(x-500)^2} = \frac{(x-5) \cdot 100}{100(x-500)^2}

Step 3: Combine the Fractions

Now that we have rewritten each fraction with the common denominator, we can combine them.

(x−395)2⋅(x−500)2100(x−500)2−(x−5)⋅100100(x−500)2\frac{(x-395)^2 \cdot (x-500)^2}{100(x-500)^2} - \frac{(x-5) \cdot 100}{100(x-500)^2}

Simplifying the Expression

Now that we have combined the fractions, we can simplify the expression by canceling out any common factors.

(x−395)2⋅(x−500)2−(x−5)⋅100100(x−500)2\frac{(x-395)^2 \cdot (x-500)^2 - (x-5) \cdot 100}{100(x-500)^2}

Expanding the Numerator

To simplify the expression further, we can expand the numerator.

(x2−790x+156025)⋅(x−500)2−(x−5)⋅100100(x−500)2\frac{(x^2 - 790x + 156025) \cdot (x-500)^2 - (x-5) \cdot 100}{100(x-500)^2}

Simplifying the Numerator

Now that we have expanded the numerator, we can simplify it by combining like terms.

x3−1290x2+395000x−156025000−100x+500100(x−500)2\frac{x^3 - 1290x^2 + 395000x - 156025000 - 100x + 500}{100(x-500)^2}

Final Simplification

After simplifying the numerator, we can simplify the expression further by canceling out any common factors.

x3−1290x2+394900x−156025500100(x−500)2\frac{x^3 - 1290x^2 + 394900x - 156025500}{100(x-500)^2}

Conclusion

Simplifying complex algebraic expressions requires a step-by-step approach, using various mathematical techniques to simplify the expression. In this article, we have simplified the given expression: (x−395)2100−x−5(x−500)2\frac{(x-395)^2}{100}-\frac{x-5}{(x-500)^2}. We have broken down the process into manageable steps, using techniques such as finding a common denominator, rewriting fractions, combining fractions, expanding the numerator, and simplifying the numerator. By following these steps, we have simplified the expression to its final form.

Future Work

Simplifying complex algebraic expressions is a crucial skill in mathematics, and there are many other expressions that can be simplified using similar techniques. Future work could involve simplifying other complex algebraic expressions, exploring new techniques for simplifying expressions, and applying these techniques to real-world problems.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Glossary

  • Algebraic expression: An expression that consists of variables, constants, and mathematical operations.
  • Common denominator: The least common multiple of two or more denominators.
  • Fraction: A way of expressing a part of a whole as a ratio of two numbers.
  • Simplifying an expression: Reducing an expression to its simplest form by combining like terms and canceling out common factors.

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Introduction

Simplifying complex algebraic expressions is a crucial skill in mathematics, particularly in calculus, algebra, and other advanced mathematical disciplines. In our previous article, we provided a step-by-step guide on simplifying the expression: (x−395)2100−x−5(x−500)2\frac{(x-395)^2}{100}-\frac{x-5}{(x-500)^2}. In this article, we will answer some frequently asked questions (FAQs) related to simplifying complex algebraic expressions.

Q&A

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

A: The first step in simplifying a complex algebraic expression is to identify the type of expression and determine the best approach to simplify it. This may involve finding a common denominator, rewriting fractions, combining fractions, expanding the numerator, and simplifying the numerator.

Q: How do I find a common denominator for two or more fractions?

A: To find a common denominator, you need to identify the least common multiple (LCM) of the two or more denominators. The LCM is the smallest number that is a multiple of all the denominators.

Q: What is the difference between a numerator and a denominator?

A: The numerator is the top part of a fraction, while the denominator is the bottom part of a fraction. The numerator is the number being divided, while the denominator is the number by which we are dividing.

Q: How do I simplify a fraction with a quadratic expression in the numerator?

A: To simplify a fraction with a quadratic expression in the numerator, you need to expand the numerator and then simplify it by combining like terms.

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

A: The final step in simplifying a complex algebraic expression is to simplify the numerator and then cancel out any common factors.

Q: Can I use a calculator to simplify complex algebraic expressions?

A: While calculators can be useful in simplifying complex algebraic expressions, it is generally recommended to simplify expressions by hand to ensure accuracy and to develop problem-solving skills.

Q: How do I know if an expression is simplified?

A: An expression is simplified when there are no like terms that can be combined and no common factors that can be canceled out.

Q: Can I simplify expressions with variables?

A: Yes, you can simplify expressions with variables by following the same steps as simplifying expressions with constants.

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

A: Some common mistakes to avoid when simplifying complex algebraic expressions include:

  • Not finding a common denominator
  • Not rewriting fractions with the common denominator
  • Not combining like terms
  • Not canceling out common factors
  • Not checking for accuracy

Conclusion

Simplifying complex algebraic expressions is a crucial skill in mathematics, and it requires a step-by-step approach. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex algebraic expressions with confidence.

Future Work

Simplifying complex algebraic expressions is a fundamental concept in mathematics, and there are many other expressions that can be simplified using similar techniques. Future work could involve simplifying other complex algebraic expressions, exploring new techniques for simplifying expressions, and applying these techniques to real-world problems.

References

  • [1] "Algebra" by Michael Artin
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Glossary

  • Algebraic expression: An expression that consists of variables, constants, and mathematical operations.
  • Common denominator: The least common multiple of two or more denominators.
  • Fraction: A way of expressing a part of a whole as a ratio of two numbers.
  • Simplifying an expression: Reducing an expression to its simplest form by combining like terms and canceling out common factors.