What Is $\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}$ In Simplest Form?A. $\frac{9+\sqrt{3x}}{3-x}$ B. $\frac{3+\sqrt{3x}}{3-x}$ C. $\frac{9+\sqrt{3x}}{3+x}$ D. $\frac{\sqrt{3}+\sqrt{x}}{3+x}$

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Introduction

In mathematics, simplifying complex fractions is an essential skill that can be applied to various problems, including algebra and calculus. A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In this article, we will focus on simplifying a specific complex fraction: 33−x\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}. We will break down the solution into manageable steps and provide a clear explanation of each step.

Step 1: Multiply by the Conjugate

To simplify the complex fraction, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 3−x\sqrt{3}-\sqrt{x} is 3+x\sqrt{3}+\sqrt{x}. By multiplying both the numerator and the denominator by the conjugate, we can eliminate the radical in the denominator.

\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}} \cdot \frac{\sqrt{3}+\sqrt{x}}{\sqrt{3}+\sqrt{x}}

Step 2: Expand and Simplify

Now, we can expand and simplify the expression by multiplying the numerator and the denominator.

\frac{\sqrt{3}(\sqrt{3}+\sqrt{x})}{(\sqrt{3}-\sqrt{x})(\sqrt{3}+\sqrt{x})}

Using the difference of squares formula, we can simplify the denominator:

(\sqrt{3}-\sqrt{x})(\sqrt{3}+\sqrt{x}) = 3 - x

So, the expression becomes:

\frac{\sqrt{3}(\sqrt{3}+\sqrt{x})}{3-x}

Step 3: Distribute and Simplify

Now, we can distribute the 3\sqrt{3} in the numerator and simplify the expression:

\frac{\sqrt{3}\sqrt{3}+\sqrt{3}\sqrt{x}}{3-x}

Using the property of radicals, we can simplify the numerator:

\sqrt{3}\sqrt{3} = 3

So, the expression becomes:

\frac{3+\sqrt{3x}}{3-x}

Conclusion

In conclusion, the complex fraction 33−x\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}} can be simplified to 3+3x3−x\frac{3+\sqrt{3x}}{3-x}. This solution involves multiplying by the conjugate, expanding and simplifying, and distributing and simplifying. By following these steps, we can simplify complex fractions and arrive at a simplified expression.

Answer

The correct answer is:

  • B. 3+3x3−x\frac{3+\sqrt{3x}}{3-x}

This answer matches the simplified expression we obtained in the previous section.

Discussion

Simplifying complex fractions is an essential skill in mathematics, and it can be applied to various problems, including algebra and calculus. In this article, we focused on simplifying the complex fraction 33−x\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}. We broke down the solution into manageable steps and provided a clear explanation of each step. By following these steps, we can simplify complex fractions and arrive at a simplified expression.

Additional Resources

For more information on simplifying complex fractions, you can refer to the following resources:

  • Khan Academy: Simplifying Complex Fractions
  • Mathway: Simplifying Complex Fractions
  • Wolfram Alpha: Simplifying Complex Fractions

These resources provide additional examples and explanations to help you understand the concept of simplifying complex fractions.

Final Thoughts

Introduction

In our previous article, we explored the concept of simplifying complex fractions and provided a step-by-step guide on how to simplify the complex fraction 33−x\frac{\sqrt{3}}{\sqrt{3}-\sqrt{x}}. In this article, we will answer some frequently asked questions (FAQs) related to 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: Why do we need to simplify complex fractions?

A: Simplifying complex fractions is essential in mathematics because it helps us to:

  • Reduce the complexity of the fraction
  • Make it easier to work with
  • Avoid errors in calculations
  • Simplify expressions and equations

Q: How do I know when to simplify a complex fraction?

A: You should simplify a complex fraction when:

  • The fraction contains radicals or fractions in the numerator or denominator
  • The fraction is difficult to work with in its current form
  • You need to simplify the fraction to solve a problem or equation

Q: What is the conjugate of a fraction?

A: The conjugate of a fraction is the fraction with the opposite sign in the numerator and denominator. For example, the conjugate of ab\frac{a}{b} is −a−b\frac{-a}{-b}.

Q: How do I multiply a fraction by its conjugate?

A: To multiply a fraction by its conjugate, you need to multiply the numerator and denominator by the conjugate. For example, to multiply ab\frac{a}{b} by its conjugate −a−b\frac{-a}{-b}, you would multiply the numerator and denominator by −a−b\frac{-a}{-b}.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

(a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2

This formula can be used to simplify expressions that contain the product of two binomials.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, you need to identify the two binomials in the expression and multiply them together. For example, to simplify the expression (a−b)(a+b)(a-b)(a+b), you would multiply the two binomials together using the difference of squares formula.

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

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

  • Not multiplying by the conjugate
  • Not expanding and simplifying the expression
  • Not distributing and simplifying the expression
  • Not checking for errors in calculations

Conclusion

Simplifying complex fractions is an essential skill in mathematics, and it can be applied to various problems, including algebra and calculus. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex fractions and arrive at a simplified expression.

Additional Resources

For more information on simplifying complex fractions, you can refer to the following resources:

  • Khan Academy: Simplifying Complex Fractions
  • Mathway: Simplifying Complex Fractions
  • Wolfram Alpha: Simplifying Complex Fractions

These resources provide additional examples and explanations to help you understand the concept of simplifying complex fractions.

Final Thoughts

Simplifying complex fractions is a crucial skill in mathematics, and it can be applied to various problems, including algebra and calculus. By following the steps outlined in this article and avoiding common mistakes, you can simplify complex fractions and arrive at a simplified expression. Remember to multiply by the conjugate, expand and simplify, and distribute and simplify to arrive at the final answer.