Simplify The Expression:$\frac{2}{\sqrt{6}-\sqrt{2}}$

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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 simplify the expression $\frac{2}{\sqrt{6}-\sqrt{2}}$ using the concept of rationalizing the denominator.

Understanding the Concept of Rationalizing the Denominator

Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate any radicals in the denominator. This process is essential in simplifying complex fractions and making them easier to work with.

Step 1: Identify the Radical in the Denominator

The given expression has a radical in the denominator, which is $\sqrt{6}-\sqrt{2}$. To rationalize the denominator, we need to eliminate this radical.

Step 2: Multiply the Numerator and Denominator by the Conjugate of the Denominator

The conjugate of the denominator is $\sqrt{6}+\sqrt{2}$. To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator.

Step 3: Simplify the Expression

After multiplying the numerator and denominator by the conjugate of the denominator, we get:

$$\frac{2}{\sqrt{6}-\sqrt{2}} \cdot \frac{\sqrt{6}+\sqrt{2}}{\sqrt{6}+\sqrt{2}}$$

Expanding the numerator and denominator, we get:

$$\frac{2(\sqrt{6}+\sqrt{2})}{(\sqrt{6})^2 - (\sqrt{2})^2}$$

Simplifying the numerator and denominator, we get:

$$\frac{2\sqrt{6}+2\sqrt{2}}{6-2}$$

Simplifying further, we get:

$$\frac{2\sqrt{6}+2\sqrt{2}}{4}$$

Step 4: Simplify the Expression Further

We can simplify the expression further by dividing both the numerator and the denominator by their greatest common factor, which is 2.

$$\frac{2\sqrt{6}+2\sqrt{2}}{4} \div 2$$

Simplifying, we get:

$$\frac{\sqrt{6}+\sqrt{2}}{2}$$

Conclusion

In this article, we simplified the expression $\frac{2}{\sqrt{6}-\sqrt{2}}$ using the concept of rationalizing the denominator. We identified the radical in the denominator, multiplied the numerator and denominator by the conjugate of the denominator, and simplified the expression further. The final simplified expression is $\frac{\sqrt{6}+\sqrt{2}}{2}$.

Frequently Asked Questions

• What is rationalizing the denominator? Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate any radicals in the denominator.
• Why is rationalizing the denominator important? Rationalizing the denominator is important because it helps to simplify complex fractions and make them easier to work with.
• How do you rationalize the denominator? To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator.

Real-World Applications

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

• Simplifying complex fractions in finance and accounting
• Solving equations in physics and engineering
• Working with complex numbers in computer science and cryptography

Tips and Tricks

• When rationalizing the denominator, make sure to multiply both the numerator and the denominator by the conjugate of the denominator.
• Use the concept of rationalizing the denominator to simplify complex fractions and make them easier to work with.
• Practice rationalizing the denominator with different expressions to become more comfortable with the concept.

References

• [1] "Rationalizing the Denominator" by Math Open Reference
• [2] "Simplifying Complex Fractions" by Khan Academy
• [3] "Rationalizing the Denominator" by Purplemath

Further Reading

• "Simplifying Complex Fractions" by Mathway
• "Rationalizing the Denominator" by IXL
• "Simplifying Complex Fractions" by Wolfram Alpha
  

Introduction

Rationalizing the denominator is a crucial step in simplifying complex fractions, especially when dealing with square roots. In this article, we will answer some frequently asked questions about rationalizing the denominator, providing you with a better understanding of this concept.

Q&A

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate any radicals in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it helps to simplify complex fractions and make them easier to work with. It also helps to eliminate any radicals in the denominator, making it easier to perform calculations.

Q: How do you rationalize the denominator?

A: To rationalize the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator is found by changing the sign of the radical in the denominator.

Q: What is the conjugate of the denominator?

A: The conjugate of the denominator is found by changing the sign of the radical in the denominator. For example, if the denominator is $\sqrt{6}-\sqrt{2}$, the conjugate is $\sqrt{6}+\sqrt{2}$.

Q: How do you find the conjugate of the denominator?

A: To find the conjugate of the denominator, you need to change the sign of the radical in the denominator. For example, if the denominator is $\sqrt{6}-\sqrt{2}$, the conjugate is $\sqrt{6}+\sqrt{2}$.

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

A: Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate any radicals in the denominator. Simplifying a fraction involves reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common factor.

Q: Can you rationalize the denominator of a fraction with a negative exponent?

A: Yes, you can rationalize the denominator of a fraction with a negative exponent. To do this, you need to multiply both the numerator and the denominator by the conjugate of the denominator.

Q: How do you rationalize the denominator of a fraction with a variable in the denominator?

A: To rationalize the denominator of a fraction with a variable in the denominator, you need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator is found by changing the sign of the variable in the denominator.

Q: Can you rationalize the denominator of a fraction with a complex number in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a complex number in the denominator. To do this, you need to multiply both the numerator and the denominator by the conjugate of the denominator.

Tips and Tricks

• When rationalizing the denominator, make sure to multiply both the numerator and the denominator by the conjugate of the denominator.
• Use the concept of rationalizing the denominator to simplify complex fractions and make them easier to work with.
• Practice rationalizing the denominator with different expressions to become more comfortable with the concept.

Real-World Applications

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

• Simplifying complex fractions in finance and accounting
• Solving equations in physics and engineering
• Working with complex numbers in computer science and cryptography

References

• [1] "Rationalizing the Denominator" by Math Open Reference
• [2] "Simplifying Complex Fractions" by Khan Academy
• [3] "Rationalizing the Denominator" by Purplemath

Further Reading

• "Simplifying Complex Fractions" by Mathway
• "Rationalizing the Denominator" by IXL
• "Simplifying Complex Fractions" by Wolfram Alpha