Which Choice Is Equivalent To The Fraction Below When $x \geq 2$?Hint: Rationalize The Denominator And Simplify.$\frac{4}{\sqrt{x}-\sqrt{x-2}}$A. $2(\sqrt{x}-\sqrt{x-2}$\]B. $-2(\sqrt{x}+\sqrt{x-2}$\]C.

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Rationalizing the Denominator: A Step-by-Step Guide to Simplifying the Given Fraction

When dealing with fractions that involve square roots in the denominator, rationalizing the denominator is a crucial step to simplify the expression. In this article, we will focus on rationalizing the denominator and simplifying the given fraction 4x−x−2\frac{4}{\sqrt{x}-\sqrt{x-2}} when x≥2x \geq 2. We will explore the different options provided and determine which one is equivalent to the given fraction.

The given fraction is 4x−x−2\frac{4}{\sqrt{x}-\sqrt{x-2}}. To rationalize the denominator, we need to get rid of the square roots in the denominator. This can be achieved by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression a−ba-b is a+ba+b.

To rationalize the denominator, we will multiply the numerator and denominator by the conjugate of the denominator, which is x+x−2\sqrt{x}+\sqrt{x-2}.

4x−x−2⋅x+x−2x+x−2\frac{4}{\sqrt{x}-\sqrt{x-2}} \cdot \frac{\sqrt{x}+\sqrt{x-2}}{\sqrt{x}+\sqrt{x-2}}

Using the distributive property, we can simplify the expression:

4(x+x−2)(x−x−2)(x+x−2)\frac{4(\sqrt{x}+\sqrt{x-2})}{(\sqrt{x}-\sqrt{x-2})(\sqrt{x}+\sqrt{x-2})}

Expanding the denominator, we get:

4(x+x−2)x−(x−2)\frac{4(\sqrt{x}+\sqrt{x-2})}{x-(x-2)}

Simplifying the denominator, we get:

4(x+x−2)2\frac{4(\sqrt{x}+\sqrt{x-2})}{2}

Simplifying further, we get:

2(x+x−2)2(\sqrt{x}+\sqrt{x-2})

Now that we have simplified the fraction, let's compare it with the options provided:

A. 2(x−x−2)2(\sqrt{x}-\sqrt{x-2}) B. −2(x+x−2)-2(\sqrt{x}+\sqrt{x-2}) C. 2(x+x−2)2(\sqrt{x}+\sqrt{x-2})

From the simplified expression, we can see that option C is equivalent to the given fraction.

In conclusion, when x≥2x \geq 2, the fraction 4x−x−2\frac{4}{\sqrt{x}-\sqrt{x-2}} is equivalent to 2(x+x−2)2(\sqrt{x}+\sqrt{x-2}). Rationalizing the denominator and simplifying the expression is a crucial step in solving this problem. By following the steps outlined in this article, we can simplify the fraction and determine which option is equivalent to the given fraction.

The final answer is option C: 2(x+x−2)2(\sqrt{x}+\sqrt{x-2}).
Frequently Asked Questions: Rationalizing the Denominator and Simplifying the Given Fraction

In our previous article, we explored the process of rationalizing the denominator and simplifying the given fraction 4x−x−2\frac{4}{\sqrt{x}-\sqrt{x-2}} when x≥2x \geq 2. We determined that the simplified expression is equivalent to 2(x+x−2)2(\sqrt{x}+\sqrt{x-2}). In this article, we will address some of the frequently asked questions related to rationalizing the denominator and simplifying the given fraction.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is the process of getting rid of the square roots in the denominator of a fraction. This is achieved by multiplying the numerator and 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 the expression and make it easier to work with. Rationalizing the denominator helps to eliminate any square roots in the denominator, which can make the expression more manageable.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression a−ba-b is a+ba+b. For example, the conjugate of x−2x-2 is x+2x+2.

Q: How do we rationalize the denominator of a fraction?

A: To rationalize the denominator of a fraction, we multiply the numerator and denominator by the conjugate of the denominator. This helps to eliminate any square roots in the denominator.

Q: What is the difference between rationalizing the denominator and simplifying the expression?

A: Rationalizing the denominator is the process of getting rid of the square roots in the denominator, while simplifying the expression involves reducing the fraction to its simplest form.

Q: Can we simplify the expression without rationalizing the denominator?

A: No, we cannot simplify the expression without rationalizing the denominator. Rationalizing the denominator is a crucial step in simplifying the expression.

Q: What is the final answer to the given problem?

A: The final answer to the given problem is 2(x+x−2)2(\sqrt{x}+\sqrt{x-2}).

In conclusion, rationalizing the denominator and simplifying the given fraction is an important concept in mathematics. By understanding the process of rationalizing the denominator and simplifying the expression, we can solve problems more efficiently and effectively. We hope that this article has addressed some of the frequently asked questions related to rationalizing the denominator and simplifying the given fraction.

For more information on rationalizing the denominator and simplifying the given fraction, please refer to the following resources:

  • Mathway: A online math problem solver that can help you solve math problems, including rationalizing the denominator and simplifying the given fraction.
  • Khan Academy: A free online learning platform that offers video lessons and practice exercises on math topics, including rationalizing the denominator and simplifying the given fraction.
  • Math Open Reference: A free online math reference book that offers detailed explanations and examples on math topics, including rationalizing the denominator and simplifying the given fraction.