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

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems more efficiently and accurately. One of the most common techniques used to simplify expressions is rationalizing the denominator. In this article, we will focus on simplifying the expression 26−2\frac{2}{\sqrt{6}-2} using this technique.

Understanding the Expression

The given expression is 26−2\frac{2}{\sqrt{6}-2}. To simplify this expression, we need to rationalize the denominator, which means we need to get rid of the square root in the denominator. The expression can be rewritten as 26−2⋅6+26+2\frac{2}{\sqrt{6}-2} \cdot \frac{\sqrt{6}+2}{\sqrt{6}+2}.

Rationalizing the Denominator

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator 6−2\sqrt{6}-2 is 6+2\sqrt{6}+2. By multiplying both the numerator and the denominator by 6+2\sqrt{6}+2, we get:

26−2⋅6+26+2=2(6+2)(6−2)(6+2)\frac{2}{\sqrt{6}-2} \cdot \frac{\sqrt{6}+2}{\sqrt{6}+2} = \frac{2(\sqrt{6}+2)}{(\sqrt{6}-2)(\sqrt{6}+2)}

Simplifying the Expression

Now, we can simplify the expression by multiplying the numerator and the denominator. The numerator becomes 26+42\sqrt{6}+4, and the denominator becomes (6)2−(2)2(\sqrt{6})^2 - (2)^2.

(6)2−(2)2=6−4=2(\sqrt{6})^2 - (2)^2 = 6 - 4 = 2

So, the expression becomes:

26+42\frac{2\sqrt{6}+4}{2}

Final Simplification

To simplify the expression further, we can divide both the numerator and the denominator by 2. This gives us:

26+42=6+2\frac{2\sqrt{6}+4}{2} = \sqrt{6}+2

Therefore, the simplified expression is 6+2\sqrt{6}+2.

Conclusion

In this article, we simplified the expression 26−2\frac{2}{\sqrt{6}-2} using the technique of rationalizing the denominator. We multiplied both the numerator and the denominator by the conjugate of the denominator and then simplified the expression further by dividing both the numerator and the denominator by 2. The final simplified expression is 6+2\sqrt{6}+2.

Tips and Tricks

  • When rationalizing the denominator, always multiply both the numerator and the denominator by the conjugate of the denominator.
  • Make sure to simplify the expression further by dividing both the numerator and the denominator by the greatest common factor.
  • Practice simplifying expressions with different types of denominators, such as square roots and fractions.

Common Mistakes

  • Failing to rationalize the denominator completely.
  • Not simplifying the expression further by dividing both the numerator and the denominator by the greatest common factor.
  • Making mistakes when multiplying the numerator and the denominator.

Real-World Applications

  • Simplifying expressions is an essential skill in mathematics, and it has many real-world applications, such as:
  • Calculating the area and perimeter of shapes.
  • Finding the volume of solids.
  • Solving problems in physics and engineering.

Final Thoughts

Simplifying expressions is a crucial skill that helps us solve problems more efficiently and accurately. By rationalizing the denominator and simplifying the expression further, we can arrive at the final answer. Remember to practice simplifying expressions with different types of denominators and to avoid common mistakes. With practice and patience, you will become proficient in simplifying expressions and solving problems in mathematics.

Introduction

In our previous article, we simplified the expression 26−2\frac{2}{\sqrt{6}-2} using the technique of rationalizing the denominator. In this article, we will answer some frequently asked questions related to simplifying expressions and rationalizing the denominator.

Q&A

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a technique used to simplify expressions by getting rid of the square root in the denominator. It involves 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 the expression and make it easier to work with. Rationalizing the denominator helps us to avoid dealing with square roots in the denominator, which can make it difficult to simplify the expression.

Q: How do I 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 obtained by changing the sign of the second term in the denominator.

Q: What is the conjugate of the denominator?

A: The conjugate of the denominator is obtained by changing the sign of the second term in the denominator. For example, if the denominator is 6−2\sqrt{6}-2, the conjugate is 6+2\sqrt{6}+2.

Q: How do I simplify the expression after rationalizing the denominator?

A: After rationalizing the denominator, you need to simplify the expression by dividing both the numerator and the denominator by the greatest common factor.

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 rationalize the denominator completely.
  • Not simplifying the expression further by dividing both the numerator and the denominator by the greatest common factor.
  • Making mistakes when multiplying the numerator and the denominator.

Q: How do I practice simplifying expressions and rationalizing the denominator?

A: You can practice simplifying expressions and rationalizing the denominator by working on problems that involve square roots and fractions. You can also try simplifying expressions with different types of denominators, such as square roots and fractions.

Q: What are some real-world applications of simplifying expressions and rationalizing the denominator?

A: Simplifying expressions and rationalizing the denominator have many real-world applications, such as:

  • Calculating the area and perimeter of shapes.
  • Finding the volume of solids.
  • Solving problems in physics and engineering.

Tips and Tricks

  • Always rationalize the denominator completely.
  • Simplify the expression further by dividing both the numerator and the denominator by the greatest common factor.
  • Practice simplifying expressions with different types of denominators, such as square roots and fractions.

Common Mistakes

  • Failing to rationalize the denominator completely.
  • Not simplifying the expression further by dividing both the numerator and the denominator by the greatest common factor.
  • Making mistakes when multiplying the numerator and the denominator.

Real-World Applications

  • Simplifying expressions and rationalizing the denominator have many real-world applications, such as:
  • Calculating the area and perimeter of shapes.
  • Finding the volume of solids.
  • Solving problems in physics and engineering.

Final Thoughts

Simplifying expressions and rationalizing the denominator are essential skills that help us solve problems more efficiently and accurately. By practicing simplifying expressions and rationalizing the denominator, you will become proficient in solving problems in mathematics and have a better understanding of the real-world applications of these skills.