The Complex Fraction in Question

Oct 13, 2025 by talynt.org 125 views

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In other words, it's a fraction that has a fraction as its numerator or denominator. Complex fractions can be simplified by multiplying the numerator and denominator by the least common multiple (LCM) of the denominators.

The Complex Fraction in Question

The complex fraction we're dealing with is:

${$ \frac{\frac{2}{x-2} - \frac{3}{x^{2-4}}{\frac{5}{x}2-4} - \frac{2}{x+2}}$}$

To simplify this fraction, we need to find the common denominator of the fractions in the numerator and the denominator.

Finding the Common Denominator

The common denominator of a set of fractions is the smallest multiple that all the denominators have in common. In this case, we have two fractions in the numerator and two fractions in the denominator.

Numerator

The first fraction in the numerator is ${$ \frac{2}{x-2}$}$, and the second fraction is ${$ \frac{3}{x^2-4}$}$. The least common multiple (LCM) of the denominators of these two fractions is ${$ (x-2)(x+2)$}$.

Denominator

The first fraction in the denominator is ${$ \frac{5}{x^2-4}$}$, and the second fraction is ${$ \frac{2}{x+2}$}$. The least common multiple (LCM) of the denominators of these two fractions is also ${$ (x+2)(x-2)$}$.

Simplifying the Complex Fraction

Now that we have found the common denominator, we can simplify the complex fraction by multiplying the numerator and denominator by the common denominator.

${$ \frac{\frac{2}{x-2} - \frac{3}{x^{2-4}}{\frac{5}{x}2-4} - \frac{2}{x+2}} = \frac{(\frac{2}{x-2} - \frac{3}{x^{2-4})(x+2)(x-2)}{(\frac{5}{x}2-4} - \frac{2}{x+2})(x+2)(x-2)}$}$

Cancelling Out Common Factors

Now that we have multiplied the numerator and denominator by the common denominator, we can cancel out common factors.

${$ \frac{(\frac{2}{x-2} - \frac{3}{x^{2-4})(x+2)(x-2)}{(\frac{5}{x}2-4} - \frac{2}{x+2})(x+2)(x-2)} = \frac{2(x+2) - 3(x-2)}{5(x+2) - 2(x-2)}$}$

Simplifying the Expression

Now that we have cancelled out common factors, we can simplify the expression.

${$ \frac{2(x+2) - 3(x-2)}{5(x+2) - 2(x-2)} = \frac{2x+4-3x+6}{5x+10-2x+4}$}$

Combining Like Terms

Now that we have simplified the expression, we can combine like terms.

${$ \frac{2x+4-3x+6}{5x+10-2x+4} = \frac{-x+10}{3x+14}$}$

The Common Denominator

The common denominator of the complex fraction is ${$ (x+2)(x-2)$}$.

Conclusion

In conclusion, the common denominator of the complex fraction ${$ \frac{\frac{2}{x-2} - \frac{3}{x^{2-4}}{\frac{5}{x}2-4} - \frac{2}{x+2}}$}$ is ${$ (x+2)(x-2)$}$.

Answer

The answer is A. ${$ (x+2)(x-2)$].

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In other words, it's a fraction that has a fraction as its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to find the least common multiple (LCM) of the denominators of the fractions in the numerator and the denominator. Then, multiply the numerator and denominator by the LCM.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.

Q: How do I find the LCM of two fractions?

A: To find the LCM of two fractions, you need to find the LCM of their denominators. Then, multiply the numerator and denominator of each fraction by the LCM.

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

A: Yes, you can simplify a complex fraction by cancelling out common factors. However, you need to make sure that the common factors are present in both the numerator and the denominator.

Q: What is the common denominator of a complex fraction?

A: The common denominator of a complex fraction is the least common multiple (LCM) of the denominators of the fractions in the numerator and the denominator.

Q: How do I know if a complex fraction can be simplified?

A: A complex fraction can be simplified if the least common multiple (LCM) of the denominators of the fractions in the numerator and the denominator is a common factor of both the numerator and the denominator.

Q: Can I simplify a complex fraction with different denominators?

A: Yes, you can simplify a complex fraction with different denominators. However, you need to find the least common multiple (LCM) of the denominators and multiply the numerator and denominator by the LCM.

Q: What is the final answer to the complex fraction [ $\frac{\frac{2}{x-2} - \frac{3}{x^2-4}}{\frac{5}{x^2-4} - \frac{2}{x+2}}\$}$ ?

A: The final answer to the complex fraction ${$ \frac{\frac{2}{x-2} - \frac{3}{x^{2-4}}{\frac{5}{x}2-4} - \frac{2}{x+2}}$}$ is ${$ \frac{-x+10}{3x+14}$}$.

Q: What is the common denominator of the complex fraction ${$ \frac{\frac{2}{x-2} - \frac{3}{x^{2-4}}{\frac{5}{x}2-4} - \frac{2}{x+2}}$}$?

A: The common denominator of the complex fraction ${$ \frac{\frac{2}{x-2} - \frac{3}{x^{2-4}}{\frac{5}{x}2-4} - \frac{2}{x+2}}$}$ is ${$ (x+2)(x-2)$}$.

Conclusion

In conclusion, complex fractions can be simplified by finding the least common multiple (LCM) of the denominators of the fractions in the numerator and the denominator, and then multiplying the numerator and denominator by the LCM. The common denominator of a complex fraction is the LCM of the denominators of the fractions in the numerator and the denominator.

Final Answer

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