What Is The Common Denominator Of \[$\frac{5}{x^2-4}-\frac{2}{x+2}\$\] In The Complex Fraction \[$\frac{\frac{2}{x-2}-\frac{3}{x^2-4}}{\frac{5}{x^2-4}-\frac{2}{x+2}}\$\]?A. \[$(x+2)(x-2)\$\]B. \[$x-2\$\]C.

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Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given problem, we have a complex fraction of the form {\frac{\frac{2}{x-2}-\frac{3}{x2-4}}{\frac{5}{x2-4}-\frac{2}{x+2}}$}$. To simplify this complex fraction, we need to find the common denominator of the fractions in the numerator and the denominator.

Finding the Common Denominator

To find the common denominator of the fractions in the numerator and the denominator, we need to first factorize the denominators of the fractions. The denominator of the first fraction in the numerator is {x^2-4$}$, which can be factorized as {(x+2)(x-2)$}$. The denominator of the second fraction in the numerator is also {x^2-4$}$, which can be factorized as {(x+2)(x-2)$}$. The denominator of the first fraction in the denominator is {x^2-4$}$, which can be factorized as {(x+2)(x-2)$}$. The denominator of the second fraction in the denominator is {x+2$}$.

Identifying the Common Denominator

From the factorization of the denominators, we can see that the common denominator of the fractions in the numerator and the denominator is {(x+2)(x-2)$}$. This is because both the numerator and the denominator have the same factor {(x+2)(x-2)$}$ in their denominators.

Simplifying the Complex Fraction

Now that we have found the common denominator, we can simplify the complex fraction by multiplying the numerator and the denominator by the common denominator. This will eliminate the fractions in the numerator and the denominator, and we will be left with a simplified fraction.

Conclusion

In conclusion, the common denominator of the complex fraction {\frac{\frac{2}{x-2}-\frac{3}{x2-4}}{\frac{5}{x2-4}-\frac{2}{x+2}}$}$ is {(x+2)(x-2)$}$. This is because both the numerator and the denominator have the same factor {(x+2)(x-2)$}$ in their denominators.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

  1. Factorize the denominators of the fractions in the numerator and the denominator.
  2. Identify the common denominator of the fractions in the numerator and the denominator.
  3. Simplify the complex fraction by multiplying the numerator and the denominator by the common denominator.

Example

Let's consider an example to illustrate the concept. Suppose we have a complex fraction of the form {\frac{\frac{1}{x+1}-\frac{2}{x2-4}}{\frac{3}{x2-4}-\frac{1}{x+1}}$}$. To simplify this complex fraction, we need to find the common denominator of the fractions in the numerator and the denominator.

Solution

To find the common denominator, we need to factorize the denominators of the fractions. The denominator of the first fraction in the numerator is {x^2-4$}$, which can be factorized as {(x+2)(x-2)$}$. The denominator of the second fraction in the numerator is also {x^2-4$}$, which can be factorized as {(x+2)(x-2)$}$. The denominator of the first fraction in the denominator is {x^2-4$}$, which can be factorized as {(x+2)(x-2)$}$. The denominator of the second fraction in the denominator is {x+1$}$.

Common Denominator

From the factorization of the denominators, we can see that the common denominator of the fractions in the numerator and the denominator is {(x+2)(x-2)$}$. This is because both the numerator and the denominator have the same factor {(x+2)(x-2)$}$ in their denominators.

Simplification

Now that we have found the common denominator, we can simplify the complex fraction by multiplying the numerator and the denominator by the common denominator. This will eliminate the fractions in the numerator and the denominator, and we will be left with a simplified fraction.

Final Answer

The final answer is {(x+2)(x-2)$}$.

Key Takeaways

  • A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.
  • To simplify a complex fraction, we need to find the common denominator of the fractions in the numerator and the denominator.
  • The common denominator is the product of the factors that appear in the denominators of the fractions.
  • Once we have found the common denominator, we can simplify the complex fraction by multiplying the numerator and the denominator by the common denominator.

Common Denominator Formula

The common denominator formula is:

{\frac{a}{b} - \frac{c}{d} = \frac{a \cdot d - c \cdot b}{b \cdot d}$}$

This formula can be used to find the common denominator of two fractions.

Simplifying Complex Fractions

To simplify a complex fraction, we need to follow these steps:

  1. Factorize the denominators of the fractions in the numerator and the denominator.
  2. Identify the common denominator of the fractions in the numerator and the denominator.
  3. Simplify the complex fraction by multiplying the numerator and the denominator by the common denominator.

Conclusion

In conclusion, the common denominator of a complex fraction is the product of the factors that appear in the denominators of the fractions. To simplify a complex fraction, we need to find the common denominator and multiply the numerator and the denominator by the common denominator.