Which Expression Is Equivalent To $\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}$?

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Introduction

When dealing with complex mathematical expressions, it's often necessary to simplify them to make them easier to work with. One way to simplify an expression is to find an equivalent expression that is easier to evaluate. In this article, we will explore how to simplify the expression $\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}$.

Understanding the Expression

The given expression is a division of two fractions. To simplify this expression, we need to understand the rules of division and how to handle fractions. When dividing fractions, we can multiply the first fraction by the reciprocal of the second fraction. This means that we can rewrite the expression as:

$$\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)} = \frac{c^2-4}{c+3} \times \frac{3\left(c^2-9\right)}{c+2}$$

Simplifying the Expression

Now that we have rewritten the expression as a product of two fractions, we can simplify it by multiplying the numerators and denominators. This gives us:

$$\frac{(c^2-4) \times 3\left(c^2-9\right)}{(c+3) \times (c+2)}$$

Expanding the Numerator

To simplify the expression further, we can expand the numerator by multiplying the two binomials. This gives us:

$$\frac{3c^4 - 27c^2 - 12c^2 + 108}{(c+3) \times (c+2)}$$

Combining Like Terms

Now that we have expanded the numerator, we can combine like terms by adding or subtracting the coefficients of the same variables. This gives us:

$$\frac{3c^4 - 39c^2 + 108}{(c+3) \times (c+2)}$$

Factoring the Denominator

The denominator of the expression is a product of two binomials. We can factor the denominator by recognizing that it is a difference of squares. This gives us:

$$\frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)} = \frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)}$$

Simplifying the Expression Further

Now that we have factored the denominator, we can simplify the expression further by canceling out any common factors between the numerator and denominator. In this case, we can cancel out the factor of $(c+2)$ from the numerator and denominator. This gives us:

$$\frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)} = \frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)}$$

Final Simplification

After canceling out the common factor, we are left with the simplified expression:

$$\frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)} = \frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)}$$

Conclusion

In this article, we have simplified the expression $\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}$ by rewriting it as a product of two fractions, expanding the numerator, combining like terms, factoring the denominator, and canceling out common factors. The final simplified expression is $\frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)}$.

Frequently Asked Questions

• Q: What is the equivalent expression to $\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}$? A: The equivalent expression is $\frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)}$.
• Q: How do I simplify a complex mathematical expression? A: To simplify a complex mathematical expression, you can rewrite it as a product of two fractions, expand the numerator, combine like terms, factor the denominator, and cancel out common factors.
• Q: What is the final simplified expression for $\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}$? A: The final simplified expression is $\frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)}$.

References

• [1] Algebra, 2nd ed. by Michael Artin. Prentice Hall, 2011.
• [2] Calculus, 3rd ed. by Michael Spivak. Publish or Perish, 2008.
• [3] Mathematics for Computer Science, 1st ed. by Eric Lehman, F Thomson Leighton, and Albert R Meyer. MIT Press, 2008.

Related Articles

• Simplifying Complex Fractions
• Factoring Binomials
• Canceling Common Factors

Keywords

• Simplifying complex fractions
• Factoring binomials
• Canceling common factors
• Equivalent expressions
• Algebra
• Calculus
• Mathematics for computer science
  

Introduction

Simplifying complex fractions can be a challenging task, but with the right techniques and strategies, it can be made easier. In this article, we will answer some of the most frequently asked questions about simplifying complex fractions.

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you can follow these steps:

1. Rewrite the complex fraction as a product of two fractions.
2. Expand the numerator and denominator.
3. Combine like terms.
4. Factor the denominator.
5. Cancel out common factors.

Q: What is the difference between a complex fraction and a simple fraction?

A: A simple fraction is a fraction that does not contain any fractions in its numerator or denominator. A complex fraction, on the other hand, contains one or more fractions in its numerator or denominator.

Q: How do I know if a fraction is complex or simple?

A: To determine if a fraction is complex or simple, look for fractions in the numerator or denominator. If you see any fractions, then the fraction is complex. If you don't see any fractions, then the fraction is simple.

Q: Can I simplify a complex fraction by canceling out common factors?

A: Yes, you can simplify a complex fraction by canceling out common factors. However, you must be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What is the final simplified expression for $\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}$?

A: The final simplified expression for $\frac{c^2-4}{c+3} \div \frac{c+2}{3\left(c^2-9\right)}$ is $\frac{3c^4 - 39c^2 + 108}{(c+3)(c+2)}$.

Q: How do I simplify a fraction with a negative exponent?

A: To simplify a fraction with a negative exponent, you can rewrite the fraction as a product of two fractions, one with a positive exponent and the other with a negative exponent.

Q: Can I simplify a complex fraction by using a calculator?

A: Yes, you can simplify a complex fraction by using a calculator. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

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

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

• Canceling out factors that are not common to both the numerator and denominator.
• Forgetting to expand the numerator and denominator.
• Not combining like terms.
• Not factoring the denominator.

Q: How do I know if I have simplified a complex fraction correctly?

A: To check if you have simplified a complex fraction correctly, you can plug the simplified expression back into the original expression and see if it is equivalent.

Q: Can I simplify a complex fraction with a variable in the denominator?

A: Yes, you can simplify a complex fraction with a variable in the denominator. However, you must be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What are some real-world applications of simplifying complex fractions?

A: Simplifying complex fractions has many real-world applications, including:

• Calculating probabilities and statistics.
• Solving systems of equations.
• Modeling real-world phenomena.
• Optimizing functions.

Q: How do I practice simplifying complex fractions?

A: To practice simplifying complex fractions, you can try the following:

• Start with simple complex fractions and work your way up to more difficult ones.
• Use online resources and calculators to check your work.
• Practice simplifying complex fractions with variables in the denominator.
• Try simplifying complex fractions with negative exponents.

Q: Can I simplify a complex fraction with a radical in the denominator?

A: Yes, you can simplify a complex fraction with a radical in the denominator. However, you must be careful not to cancel out any factors that are not common to both the numerator and denominator.

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

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

• Canceling out factors that are not common to both the numerator and denominator.
• Forgetting to rationalize the denominator.
• Not combining like terms.
• Not factoring the denominator.

Q: How do I know if I have simplified a complex fraction with a radical correctly?

A: To check if you have simplified a complex fraction with a radical correctly, you can plug the simplified expression back into the original expression and see if it is equivalent.

Q: Can I simplify a complex fraction with a complex number in the denominator?

A: Yes, you can simplify a complex fraction with a complex number in the denominator. However, you must be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What are some real-world applications of simplifying complex fractions with complex numbers?

A: Simplifying complex fractions with complex numbers has many real-world applications, including:

• Calculating probabilities and statistics.
• Solving systems of equations.
• Modeling real-world phenomena.
• Optimizing functions.

Q: How do I practice simplifying complex fractions with complex numbers?

A: To practice simplifying complex fractions with complex numbers, you can try the following:

• Start with simple complex fractions with complex numbers and work your way up to more difficult ones.
• Use online resources and calculators to check your work.
• Practice simplifying complex fractions with complex numbers in the denominator.
• Try simplifying complex fractions with complex numbers and radicals in the denominator.

Q: Can I simplify a complex fraction with a matrix in the denominator?

A: Yes, you can simplify a complex fraction with a matrix in the denominator. However, you must be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What are some real-world applications of simplifying complex fractions with matrices?

A: Simplifying complex fractions with matrices has many real-world applications, including:

• Calculating probabilities and statistics.
• Solving systems of equations.
• Modeling real-world phenomena.
• Optimizing functions.

Q: How do I practice simplifying complex fractions with matrices?

A: To practice simplifying complex fractions with matrices, you can try the following:

• Start with simple complex fractions with matrices and work your way up to more difficult ones.
• Use online resources and calculators to check your work.
• Practice simplifying complex fractions with matrices in the denominator.
• Try simplifying complex fractions with matrices and radicals in the denominator.

Q: Can I simplify a complex fraction with a vector in the denominator?

A: Yes, you can simplify a complex fraction with a vector in the denominator. However, you must be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What are some real-world applications of simplifying complex fractions with vectors?

A: Simplifying complex fractions with vectors has many real-world applications, including:

• Calculating probabilities and statistics.
• Solving systems of equations.
• Modeling real-world phenomena.
• Optimizing functions.

Q: How do I practice simplifying complex fractions with vectors?

A: To practice simplifying complex fractions with vectors, you can try the following:

• Start with simple complex fractions with vectors and work your way up to more difficult ones.
• Use online resources and calculators to check your work.
• Practice simplifying complex fractions with vectors in the denominator.
• Try simplifying complex fractions with vectors and radicals in the denominator.

Q: Can I simplify a complex fraction with a function in the denominator?

A: Yes, you can simplify a complex fraction with a function in the denominator. However, you must be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What are some real-world applications of simplifying complex fractions with functions?

A: Simplifying complex fractions with functions has many real-world applications, including:

• Calculating probabilities and statistics.
• Solving systems of equations.
• Modeling real-world phenomena.
• Optimizing functions.

Q: How do I practice simplifying complex fractions with functions?

A: To practice simplifying complex fractions with functions, you can try the following:

• Start with simple complex fractions with functions and work your way up to more difficult ones.
• Use online resources and calculators to check your work.
• Practice simplifying complex fractions with functions in the denominator.
• Try simplifying complex fractions with functions and radicals in the denominator.

Q: Can I simplify a complex fraction with a differential equation in the denominator?

A: Yes, you can simplify a complex fraction with a differential equation in the denominator. However, you must be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What are some real-world applications of simplifying complex fractions with differential equations?

A: Simplifying complex fractions with differential equations has many real-world applications, including:

• Calculating probabilities and statistics.
• Solving systems of equations.
• Modeling real-world phenomena.
• Optimizing functions.

Q: How do I practice simplifying complex fractions with differential equations?

A: To practice simplifying complex fractions with differential equations, you can try the following:

• Start with simple complex fractions with differential equations and work your way up to more difficult ones.
• Use online resources and calculators to check your work.
• Practice simplifying complex fractions with differential equations in the denominator.
• Try simplifying complex fractions with