Rationalizing the First Denominator

Oct 13, 2025 by wbrum.org 91 views

=====================================================

Introduction

In this article, we will delve into simplifying a given expression involving square roots. The expression is $\frac{\sqrt{2}}{\sqrt{3}-\sqrt{2}}-\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ . We will break down the solution into manageable steps, making it easier to understand and follow along.

Step 1: Rationalize the Denominators

To simplify the given expression, we need to rationalize the denominators of both fractions. Rationalizing the denominator involves multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression $a-b$ is $a+b$ .

Rationalizing the First Denominator

The first denominator is $\sqrt{3}-\sqrt{2}$ . To rationalize this denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{3}+\sqrt{2}$ .

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

This simplifies to:

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

Using the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$ , we can simplify the denominator:

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

So, the first fraction simplifies to:

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

Rationalizing the Second Denominator

The second denominator is $\sqrt{3}+\sqrt{2}$ . To rationalize this denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{3}-\sqrt{2}$ .

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

This simplifies to:

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

Using the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$ , we can simplify the denominator:

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

So, the second fraction simplifies to:

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

Step 2: Simplify the Expression

Now that we have rationalized the denominators, we can simplify the expression by combining the two fractions.

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

To subtract these fractions, we need to have a common denominator, which is 1 in this case. So, we can simply subtract the numerators:

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

Expanding the squared term, we get:

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

Simplifying further, we get:

\sqrt{2}(\sqrt{3}+\sqrt{2}) - (3 - 2\sqrt{6} + 2)

Combining like terms, we get:

\sqrt{2}(\sqrt{3}+\sqrt{2}) - 1 - 2\sqrt{6}

Step 3: Final Simplification

To simplify the expression further, we can expand the first term:

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

Using the property of square roots that $\sqrt{a}\sqrt{b} = \sqrt{ab}$ , we can simplify the first term:

\sqrt{2}\sqrt{3} + \sqrt{2}\sqrt{2} = \sqrt{6} + 2

So, the final simplified expression is:

\sqrt{6} + 2 - 1 - 2\sqrt{6}

Combining like terms, we get:

\sqrt{6} + 1 - 2\sqrt{6}

Simplifying further, we get:

- \sqrt{6} + 1

The final answer is $\boxed{-\sqrt{6} + 1}$ .

Conclusion

In this article, we simplified the given expression involving square roots. We rationalized the denominators, combined the fractions, and simplified the expression further. The final simplified expression is $\boxed{-\sqrt{6} + 1}$ .

=====================================================

Introduction

In our previous article, we simplified the given expression involving square roots. We rationalized the denominators, combined the fractions, and simplified the expression further. In this article, we will provide a Q&A guide to help you understand the solution better.

Q&A

Q: What is the given expression?

A: The given expression is $\frac{\sqrt{2}}{\sqrt{3}-\sqrt{2}}-\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ .

Q: Why do we need to rationalize the denominators?

A: We need to rationalize the denominators to simplify the expression. Rationalizing the denominator involves multiplying the numerator and denominator by the conjugate of the denominator.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a-b$ is $a+b$ .

Q: How do we rationalize the first denominator?

A: To rationalize the first denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{3}+\sqrt{2}$ .

Q: How do we simplify the first fraction after rationalizing the denominator?

A: After rationalizing the denominator, we simplify the first fraction by multiplying the numerator and denominator by the conjugate of the denominator. This results in $\sqrt{2}(\sqrt{3}+\sqrt{2})$ .

Q: How do we rationalize the second denominator?

A: To rationalize the second denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is $\sqrt{3}-\sqrt{2}$ .

Q: How do we simplify the second fraction after rationalizing the denominator?

A: After rationalizing the denominator, we simplify the second fraction by multiplying the numerator and denominator by the conjugate of the denominator. This results in $(\sqrt{3}-\sqrt{2})^2$ .

Q: How do we combine the two fractions?

A: We combine the two fractions by subtracting the second fraction from the first fraction.

Q: What is the final simplified expression?

A: The final simplified expression is $\boxed{-\sqrt{6} + 1}$ .

Common Mistakes to Avoid

Mistake 1: Not Rationalizing the Denominators

Not rationalizing the denominators can lead to a complex and difficult-to-simplify expression.

Mistake 2: Not Using the Conjugate of the Denominator

Not using the conjugate of the denominator can lead to an incorrect simplification of the expression.

Mistake 3: Not Combining Like Terms

Not combining like terms can lead to a complex and difficult-to-simplify expression.

Conclusion

In this article, we provided a Q&A guide to help you understand the solution to the given expression involving square roots. We covered common mistakes to avoid and provided a step-by-step guide to simplifying the expression. By following these steps and avoiding common mistakes, you can simplify the given expression and arrive at the final answer.

Final Answer

The final answer is $\boxed{-\sqrt{6} + 1}$ .