Simplify The Expression:$ \left(\frac{a}{b} - \frac{b}{a}\right)\left(a - \frac{a^2}{a+b}\right) }$Choose The Correct Option A. { \frac{a-b {a}$}$B. { A-b$}$C. 0

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Introduction

In this article, we will simplify the given expression step by step. The expression is a product of two fractions, and we will use algebraic manipulation to simplify it. We will start by simplifying each fraction separately and then multiply them together.

Step 1: Simplify the First Fraction

The first fraction is (ab−ba)\left(\frac{a}{b} - \frac{b}{a}\right). To simplify this fraction, we need to find a common denominator. The common denominator of aa and bb is abab. Therefore, we can rewrite the fraction as:

(a2−b2ab)\left(\frac{a^2 - b^2}{ab}\right)

Step 2: Simplify the Second Fraction

The second fraction is (a−a2a+b)\left(a - \frac{a^2}{a+b}\right). To simplify this fraction, we need to find a common denominator. The common denominator of aa and a+ba+b is a+ba+b. Therefore, we can rewrite the fraction as:

(a(a+b)−a2a+b)\left(\frac{a(a+b) - a^2}{a+b}\right)

Step 3: Simplify the Second Fraction Further

We can simplify the numerator of the second fraction further by combining like terms:

(a2+ab−a2a+b)\left(\frac{a^2 + ab - a^2}{a+b}\right)

Step 4: Simplify the Second Fraction Even Further

We can simplify the numerator of the second fraction even further by canceling out the a2a^2 term:

(aba+b)\left(\frac{ab}{a+b}\right)

Step 5: Multiply the Two Fractions Together

Now that we have simplified both fractions, we can multiply them together:

(a2−b2ab)(aba+b)\left(\frac{a^2 - b^2}{ab}\right)\left(\frac{ab}{a+b}\right)

Step 6: Simplify the Product

We can simplify the product by canceling out the abab term:

(a2−b2a+b)\left(\frac{a^2 - b^2}{a+b}\right)

Step 7: Factor the Numerator

We can factor the numerator of the fraction by recognizing that it is a difference of squares:

((a−b)(a+b)a+b)\left(\frac{(a-b)(a+b)}{a+b}\right)

Step 8: Cancel Out the Common Term

We can cancel out the common term a+ba+b in the numerator and denominator:

a−ba-b

Conclusion

We have simplified the given expression step by step. The final simplified expression is a−ba-b. This is the correct answer.

Discussion

The given expression is a product of two fractions. We simplified each fraction separately and then multiplied them together. We used algebraic manipulation to simplify the expression, including factoring and canceling out common terms.

Final Answer

The final answer is a−b\boxed{a-b}.

Related Topics

  • Simplifying algebraic expressions
  • Factoring and canceling out common terms
  • Algebraic manipulation

References

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Introduction

In our previous article, we simplified the given expression step by step. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

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

A: The first step in simplifying the expression is to simplify each fraction separately. We can start by finding a common denominator for the fractions.

Q: How do we simplify the first fraction?

A: To simplify the first fraction, we need to find a common denominator. The common denominator of aa and bb is abab. Therefore, we can rewrite the fraction as:

(a2−b2ab)\left(\frac{a^2 - b^2}{ab}\right)

Q: How do we simplify the second fraction?

A: To simplify the second fraction, we need to find a common denominator. The common denominator of aa and a+ba+b is a+ba+b. Therefore, we can rewrite the fraction as:

(a(a+b)−a2a+b)\left(\frac{a(a+b) - a^2}{a+b}\right)

Q: What is the final simplified expression?

A: The final simplified expression is a−ba-b.

Q: Why do we need to factor the numerator?

A: We need to factor the numerator because it is a difference of squares. Factoring the numerator allows us to simplify the expression further.

Q: Why do we need to cancel out the common term?

A: We need to cancel out the common term because it appears in both the numerator and denominator. Canceling out the common term allows us to simplify the expression further.

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

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

  • Not finding a common denominator
  • Not factoring the numerator
  • Not canceling out common terms
  • Not simplifying the expression step by step

Tips and Tricks

  • Always start by simplifying each fraction separately.
  • Use a common denominator to simplify fractions.
  • Factor the numerator to simplify the expression further.
  • Cancel out common terms to simplify the expression further.
  • Simplify the expression step by step.

Conclusion

Simplifying expressions can be a challenging task, but with practice and patience, you can become proficient in simplifying expressions. Remember to always start by simplifying each fraction separately, use a common denominator, factor the numerator, cancel out common terms, and simplify the expression step by step.

Final Answer

The final answer is a−b\boxed{a-b}.

Related Topics

  • Simplifying algebraic expressions
  • Factoring and canceling out common terms
  • Algebraic manipulation

References