What Is The Simplest Form Of $\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}$?A. $2 \sqrt{6} + 4$ B. $2 \sqrt{5} + 4$ C. $\frac{2 \sqrt{6} + 4}{5}$ D. $\frac{2 \sqrt{5} + 4}{5}$

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**What is the Simplest Form of $\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}$?**

Understanding the Problem

The given expression is a fraction that involves square roots. To simplify this expression, we need to rationalize the denominator, which means removing the square root from the denominator.

Step 1: Multiply by the Conjugate

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 3−2\sqrt{3}-\sqrt{2} is 3+2\sqrt{3}+\sqrt{2}.

\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} \cdot \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}

Step 2: Simplify the Expression

Now, we can simplify the expression by multiplying the numerators and the denominators.

\frac{2 \sqrt{2}(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}

Step 3: Use the Difference of Squares Formula

The denominator can be simplified using the difference of squares formula: (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2.

(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}) = \sqrt{3}^2 - \sqrt{2}^2 = 3 - 2 = 1

Step 4: Simplify the Expression

Now, we can simplify the expression by substituting the simplified denominator.

\frac{2 \sqrt{2}(\sqrt{3}+\sqrt{2})}{1} = 2 \sqrt{2}(\sqrt{3}+\sqrt{2})

Step 5: Simplify the Expression Further

We can simplify the expression further by multiplying the terms.

2 \sqrt{2}(\sqrt{3}+\sqrt{2}) = 2 \sqrt{6} + 4

Conclusion

The simplest form of 223−2\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} is 26+42 \sqrt{6} + 4.

Q&A

Q: What is the simplest form of 223−2\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}}?

A: The simplest form of 223−2\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} is 26+42 \sqrt{6} + 4.

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.

Q: What is the conjugate of 3−2\sqrt{3}-\sqrt{2}?

A: The conjugate of 3−2\sqrt{3}-\sqrt{2} is 3+2\sqrt{3}+\sqrt{2}.

Q: How do I simplify the expression?

A: To simplify the expression, you need to multiply the numerators and the denominators, and then use the difference of squares formula to simplify the denominator.

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.

Q: How do I simplify the expression further?

A: To simplify the expression further, you need to multiply the terms.

Q: What is the final answer?

A: The final answer is 26+42 \sqrt{6} + 4.

Common Mistakes

  • Not rationalizing the denominator
  • Not using the conjugate to simplify the expression
  • Not using the difference of squares formula to simplify the denominator
  • Not multiplying the terms to simplify the expression further

Tips and Tricks

  • Make sure to rationalize the denominator to simplify the expression.
  • Use the conjugate to simplify the expression.
  • Use the difference of squares formula to simplify the denominator.
  • Multiply the terms to simplify the expression further.

Conclusion

In conclusion, the simplest form of 223−2\frac{2 \sqrt{2}}{\sqrt{3}-\sqrt{2}} is 26+42 \sqrt{6} + 4. To simplify this expression, you need to rationalize the denominator, use the conjugate, use the difference of squares formula, and multiply the terms.