Which Choice Is Equivalent To The Fraction Below When $x \geq 1$?Rationalize The Denominator And Simplify: $\frac{1}{\sqrt{x}-\sqrt{x-1}}$A. $\sqrt{x}+\sqrt{x-1}$B. $\sqrt{x}-\sqrt{x-1}$C.

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Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. In this article, we will explore how to rationalize the denominator of the given fraction 1xβˆ’xβˆ’1\frac{1}{\sqrt{x}-\sqrt{x-1}} and simplify it. We will also examine the different choices provided and determine which one is equivalent to the given fraction.

Understanding the Problem

The given fraction is 1xβˆ’xβˆ’1\frac{1}{\sqrt{x}-\sqrt{x-1}}. To rationalize the denominator, we need to eliminate the square roots in the denominator. This can be achieved by multiplying both the numerator and the denominator by the conjugate of the denominator.

Rationalizing the Denominator

The conjugate of the denominator xβˆ’xβˆ’1\sqrt{x}-\sqrt{x-1} is x+xβˆ’1\sqrt{x}+\sqrt{x-1}. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate:

1xβˆ’xβˆ’1β‹…x+xβˆ’1x+xβˆ’1\frac{1}{\sqrt{x}-\sqrt{x-1}} \cdot \frac{\sqrt{x}+\sqrt{x-1}}{\sqrt{x}+\sqrt{x-1}}

This simplifies to:

x+xβˆ’1(xβˆ’xβˆ’1)(x+xβˆ’1)\frac{\sqrt{x}+\sqrt{x-1}}{(\sqrt{x}-\sqrt{x-1})(\sqrt{x}+\sqrt{x-1})}

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

(xβˆ’xβˆ’1)(x+xβˆ’1)=xβˆ’(xβˆ’1)=1(\sqrt{x}-\sqrt{x-1})(\sqrt{x}+\sqrt{x-1}) = x - (x-1) = 1

Therefore, the fraction simplifies to:

x+xβˆ’1\sqrt{x}+\sqrt{x-1}

Analyzing the Choices

Now that we have simplified the fraction, let's examine the different choices provided:

A. x+xβˆ’1\sqrt{x}+\sqrt{x-1} B. xβˆ’xβˆ’1\sqrt{x}-\sqrt{x-1} C. x+xβˆ’1βˆ’x+xβˆ’1\sqrt{x}+\sqrt{x-1} - \sqrt{x}+\sqrt{x-1}

Based on our simplification, we can see that choice A is equivalent to the given fraction.

Conclusion

In conclusion, rationalizing the denominator of the given fraction 1xβˆ’xβˆ’1\frac{1}{\sqrt{x}-\sqrt{x-1}} and simplifying it results in x+xβˆ’1\sqrt{x}+\sqrt{x-1}. This is equivalent to choice A. Therefore, the correct answer is A.

Additional Tips and Tricks

When rationalizing the denominator, it's essential to remember the following tips and tricks:

  • Multiply both the numerator and the denominator by the conjugate of the denominator.
  • Use the difference of squares formula to simplify the denominator.
  • Be careful when simplifying the fraction, as it's easy to make mistakes.

By following these tips and tricks, you can confidently rationalize the denominator and simplify complex fractions.

Common Mistakes to Avoid

When rationalizing the denominator, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Failing to multiply both the numerator and the denominator by the conjugate of the denominator.
  • Not using the difference of squares formula to simplify the denominator.
  • Making mistakes when simplifying the fraction.

By avoiding these common mistakes, you can ensure that your rationalization is accurate and correct.

Real-World Applications

Rationalizing the denominator has numerous real-world applications, including:

  • Simplifying complex fractions in algebra and calculus.
  • Solving equations and inequalities involving square roots.
  • Working with rational expressions and functions.

By mastering the art of rationalizing the denominator, you can tackle complex problems with confidence and accuracy.

Conclusion

Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. In this article, we will explore some frequently asked questions about rationalizing the denominator and provide detailed answers.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is the process of eliminating the square roots in the denominator of a fraction. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify complex fractions and make them easier to work with. Rationalizing the denominator helps to eliminate the square roots in the denominator, making it easier to perform operations such as addition, subtraction, multiplication, and division.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, follow these steps:

  1. Multiply both the numerator and the denominator by the conjugate of the denominator.
  2. Use the difference of squares formula to simplify the denominator.
  3. Be careful when simplifying the fraction, as it's easy to make mistakes.

Q: What is the conjugate of the denominator?

A: The conjugate of the denominator is the expression obtained by changing the sign of the second term in the denominator. For example, if the denominator is xβˆ’xβˆ’1\sqrt{x}-\sqrt{x-1}, the conjugate is x+xβˆ’1\sqrt{x}+\sqrt{x-1}.

Q: How do I simplify the denominator using the difference of squares formula?

A: The difference of squares formula states that (aβˆ’b)(a+b)=a2βˆ’b2(a-b)(a+b) = a^2 - b^2. To simplify the denominator, multiply the two binomials using the difference of squares formula:

(xβˆ’xβˆ’1)(x+xβˆ’1)=xβˆ’(xβˆ’1)=1(\sqrt{x}-\sqrt{x-1})(\sqrt{x}+\sqrt{x-1}) = x - (x-1) = 1

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

  • Failing to multiply both the numerator and the denominator by the conjugate of the denominator.
  • Not using the difference of squares formula to simplify the denominator.
  • Making mistakes when simplifying the fraction.

Q: How do I apply rationalizing the denominator in real-world problems?

A: Rationalizing the denominator has numerous real-world applications, including:

  • Simplifying complex fractions in algebra and calculus.
  • Solving equations and inequalities involving square roots.
  • Working with rational expressions and functions.

By mastering the art of rationalizing the denominator, you can tackle complex problems with confidence and accuracy.

Q: What are some tips and tricks for rationalizing the denominator?

A: Here are some tips and tricks for rationalizing the denominator:

  • Multiply both the numerator and the denominator by the conjugate of the denominator.
  • Use the difference of squares formula to simplify the denominator.
  • Be careful when simplifying the fraction, as it's easy to make mistakes.

By following these tips and tricks, you can confidently rationalize the denominator and simplify complex fractions.

Conclusion

In conclusion, rationalizing the denominator is a crucial step in simplifying complex fractions. By following the steps outlined in this article, you can confidently rationalize the denominator and simplify the given fraction. Remember to multiply both the numerator and the denominator by the conjugate of the denominator, use the difference of squares formula to simplify the denominator, and be careful when simplifying the fraction. With practice and patience, you can master the art of rationalizing the denominator and tackle complex problems with confidence and accuracy.