What Is The Simplest Form Of 2 3 6 \frac{2 \sqrt{3}}{\sqrt{6}} 6 ​ 2 3 ​ ​ ?A. 2 \sqrt{2} 2 ​ B. 3 \sqrt{3} 3 ​ C. 3 2 3 \sqrt{2} 3 2 ​ D. 2 3 2 \sqrt{3} 2 3 ​

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Introduction

Rationalizing the denominator and simplifying expressions involving square roots are essential skills in mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying the expression $\frac{2 \sqrt{3}}{\sqrt{6}}$ to its simplest form. This involves understanding the properties of square roots, rationalizing the denominator, and simplifying the resulting expression.

Understanding Square Roots

Before we proceed with simplifying the expression, it's essential to understand the properties of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. We can represent square roots using the symbol $\sqrt{}$. For instance, $\sqrt{16}$ represents the square root of 16.

Simplifying the Expression

To simplify the expression $\frac{2 \sqrt{3}}{\sqrt{6}}$, we need to rationalize the denominator. Rationalizing the denominator involves getting rid of any square roots in the denominator. We can do this by multiplying both the numerator and the denominator by the square root of the number in the denominator.

Step 1: Multiply the Numerator and Denominator by $\sqrt{6}$

To rationalize the denominator, we need to multiply both the numerator and the denominator by $\sqrt{6}$. This will eliminate the square root in the denominator.

$\frac{2 \sqrt{3}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 3: Simplify the Expression

Now that we have multiplied the numerator and the denominator by $\sqrt{6}$, we can simplify the expression.

$\frac{2 \sqrt{3} \cdot \sqrt{6}}{\sqrt{6} \cdot \sqrt{6}}$

Step 4: Simplify the Square Roots

We can simplify the square roots in the numerator and the denominator.

$\frac{2 \sqrt{18}}{6}$

Step 5: Simplify the Square Root of 18

The square root of 18 can be simplified further.

$\sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2}$

Step 6: Substitute the Simplified Square Root

Now that we have simplified the square root of 18, we can substitute it back into the expression.

$\frac{2 \cdot 3 \sqrt{2}}{6}$

Step 7: Simplify the Expression

We can simplify the expression further by canceling out any common factors.

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

Step 8: Final Simplification

The final simplification of the expression is:

$\sqrt{2}$

Conclusion

In this article, we simplified the expression $\frac{2 \sqrt{3}}{\sqrt{6}}$ to its simplest form. We used the properties of square roots, rationalized the denominator, and simplified the resulting expression. The final simplified form of the expression is $\sqrt{2}$.

Final Answer

The final answer is $\boxed{\sqrt{2}}$.

Introduction

In our previous article, we simplified the expression $\frac{2 \sqrt{3}}{\sqrt{6}}$ to its simplest form. In this article, we will address some frequently asked questions (FAQs) about simplifying expressions with square roots. These FAQs will provide additional insights and examples to help you better understand the process of simplifying expressions involving square roots.

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be simplified to a rational number, which is a number that can be expressed as the ratio of two integers. An irrational expression, on the other hand, is an expression that cannot be simplified to a rational number. Expressions involving square roots are often irrational.

Q: How do I simplify an expression with a square root in the denominator?

A: To simplify an expression with a square root in the denominator, you need to rationalize the denominator. This involves multiplying both the numerator and the denominator by the square root of the number in the denominator.

Q: What is the property of square roots that allows us to simplify expressions?

A: The property of square roots that allows us to simplify expressions is the fact that the square root of a product is equal to the product of the square roots. For example, $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$.

Q: How do I simplify an expression with multiple square roots in the denominator?

A: To simplify an expression with multiple square roots in the denominator, you need to rationalize the denominator by multiplying both the numerator and the denominator by the product of the square roots in the denominator.

Q: What is the difference between a perfect square and an imperfect square?

A: A perfect square is a number that can be expressed as the square of an integer. For example, 16 is a perfect square because it can be expressed as $4^2$. An imperfect square, on the other hand, is a number that cannot be expressed as the square of an integer.

Q: How do I simplify an expression with a square root in the numerator and a perfect square in the denominator?

A: To simplify an expression with a square root in the numerator and a perfect square in the denominator, you need to cancel out the perfect square in the denominator.

Q: What is the final simplified form of the expression $\frac{2 \sqrt{3}}{\sqrt{6}}$?

A: The final simplified form of the expression $\frac{2 \sqrt{3}}{\sqrt{6}}$ is $\sqrt{2}$.

Q: Can I simplify an expression with a square root in the denominator if the square root is not a perfect square?

A: Yes, you can simplify an expression with a square root in the denominator even if the square root is not a perfect square. You need to rationalize the denominator by multiplying both the numerator and the denominator by the square root of the number in the denominator.

Q: How do I know if an expression is already in its simplest form?

A: To determine if an expression is already in its simplest form, you need to check if the expression cannot be simplified further by canceling out any common factors or by rationalizing the denominator.

Conclusion

In this article, we addressed some frequently asked questions (FAQs) about simplifying expressions with square roots. These FAQs provided additional insights and examples to help you better understand the process of simplifying expressions involving square roots. By following the steps outlined in this article, you can simplify expressions with square roots and arrive at their simplest forms.

Final Answer

The final answer is $\boxed{\sqrt{2}}$.