Simplify The Following Expression:$ \frac{\frac{a 3-27}{a 2-9}}{\frac{a^2+3a+9}{a+3}} }$Options A. 1B. { \frac{-1 {a+3}$}$C. { A+3$}$

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Introduction

In this article, we will simplify the given expression using algebraic manipulation. The expression is a complex fraction, and we will break it down into smaller parts to simplify it. We will use various algebraic techniques, such as factoring and canceling, to simplify the expression.

The Given Expression

The given expression is:

a3−27a2−9a2+3a+9a+3\frac{\frac{a^3-27}{a^2-9}}{\frac{a^2+3a+9}{a+3}}

Step 1: Factor the Numerator and Denominator

We can start by factoring the numerator and denominator of the expression. The numerator can be factored as:

a3−27=(a−3)(a2+3a+9)a^3-27 = (a-3)(a^2+3a+9)

The denominator can be factored as:

a2−9=(a−3)(a+3)a^2-9 = (a-3)(a+3)

Step 2: Simplify the Expression

Now that we have factored the numerator and denominator, we can simplify the expression. We can cancel out the common factor of (a−3)(a-3) from the numerator and denominator:

(a−3)(a2+3a+9)(a−3)(a+3)a2+3a+9a+3\frac{\frac{(a-3)(a^2+3a+9)}{(a-3)(a+3)}}{\frac{a^2+3a+9}{a+3}}

This simplifies to:

a2+3a+9a+3\frac{a^2+3a+9}{a+3}

Step 3: Factor the Quadratic Expression

The quadratic expression a2+3a+9a^2+3a+9 can be factored as:

(a+3)2(a+3)^2

Step 4: Simplify the Expression Further

Now that we have factored the quadratic expression, we can simplify the expression further. We can cancel out the common factor of (a+3)(a+3) from the numerator and denominator:

(a+3)2a+3\frac{(a+3)^2}{a+3}

This simplifies to:

a+3a+3

Conclusion

In this article, we simplified the given expression using algebraic manipulation. We factored the numerator and denominator, canceled out common factors, and simplified the expression further. The final simplified expression is:

a+3a+3

This is the correct answer.

Final Answer

The final answer is:

a+3\boxed{a+3}

Discussion

The given expression is a complex fraction, and we simplified it using various algebraic techniques. We factored the numerator and denominator, canceled out common factors, and simplified the expression further. The final simplified expression is a+3a+3. This is the correct answer.

Related Topics

  • Simplifying complex fractions
  • Factoring quadratic expressions
  • Canceling common factors

References

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Introduction

In our previous article, we simplified the given expression using algebraic manipulation. We factored the numerator and denominator, canceled out common factors, and simplified the expression further. In this article, we will answer some frequently asked questions related to the simplification of the given expression.

Q&A

Q1: What is the given expression?

A1: The given expression is:

a3−27a2−9a2+3a+9a+3\frac{\frac{a^3-27}{a^2-9}}{\frac{a^2+3a+9}{a+3}}

Q2: How do we simplify the given expression?

A2: We simplify the given expression by factoring the numerator and denominator, canceling out common factors, and simplifying the expression further.

Q3: What is the final simplified expression?

A3: The final simplified expression is:

a+3a+3

Q4: Why do we need to factor the numerator and denominator?

A4: We need to factor the numerator and denominator to cancel out common factors and simplify the expression further.

Q5: What is the importance of canceling common factors?

A5: Canceling common factors is important because it helps to simplify the expression and make it easier to work with.

Q6: Can we simplify the expression further?

A6: Yes, we can simplify the expression further by factoring the quadratic expression and canceling out common factors.

Q7: What is the final answer?

A7: The final answer is:

a+3\boxed{a+3}

Additional Tips and Tricks

  • When simplifying complex fractions, it's essential to factor the numerator and denominator and cancel out common factors.
  • Factoring quadratic expressions can help to simplify the expression further.
  • Canceling common factors is a crucial step in simplifying complex fractions.

Common Mistakes to Avoid

  • Not factoring the numerator and denominator can lead to incorrect simplification.
  • Not canceling common factors can result in a more complex expression.
  • Not simplifying the expression further can lead to a more complicated solution.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the given expression. We provided additional tips and tricks and common mistakes to avoid. We hope that this article has been helpful in understanding the simplification of complex fractions.

Final Answer

The final answer is:

a+3\boxed{a+3}

Discussion

The given expression is a complex fraction, and we simplified it using various algebraic techniques. We factored the numerator and denominator, canceled out common factors, and simplified the expression further. The final simplified expression is a+3a+3. This is the correct answer.

Related Topics

  • Simplifying complex fractions
  • Factoring quadratic expressions
  • Canceling common factors

References