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

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Introduction

Rationalizing the denominator and simplifying expressions with 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. We will use various mathematical techniques, including rationalizing the denominator and simplifying expressions with square roots.

Understanding the Expression

The given expression is $\frac{2 \sqrt{3}}{\sqrt{6}}$. To simplify this expression, we need to rationalize the denominator, which means removing the square root from the denominator. We can do this by multiplying both the numerator and the denominator by the square root of the denominator.

Rationalizing the Denominator

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

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

Simplifying the Expression

Now that we have rationalized the denominator, we can simplify the expression. We can start by simplifying the numerator.

$\sqrt{3} \cdot \sqrt{6} = \sqrt{3 \cdot 6} = \sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2}$

Simplifying the Denominator

The denominator is $\sqrt{6} \cdot \sqrt{6} = 6$.

Final Simplification

Now that we have simplified the numerator and the denominator, we can simplify the expression.

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

Conclusion

In this article, we simplified the expression $\frac{2 \sqrt{3}}{\sqrt{6}}$ to its simplest form, which is $\sqrt{2}$. We used various mathematical techniques, including rationalizing the denominator and simplifying expressions with square roots. This expression is a fundamental concept in mathematics, and understanding how to simplify it is essential for solving problems in algebra and geometry.

Final Answer

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

Comparison of Options

Let's compare the final answer with the given options.

A. $\sqrt{2}$ - This is the correct answer. B. $\sqrt{3}$ - This is not the correct answer. C. $3 \sqrt{2}$ - This is not the correct answer. D. $2 \sqrt{3}$ - This is not the correct answer.

Importance of Simplifying Expressions

Simplifying expressions with square roots is an essential skill in mathematics. It helps us to:

• Understand the underlying structure of the expression
• Identify patterns and relationships between different parts of the expression
• Solve problems more efficiently and effectively
• Develop critical thinking and problem-solving skills

Tips for Simplifying Expressions

Here are some tips for simplifying expressions with square roots:

• Rationalize the denominator by multiplying both the numerator and the denominator by the square root of the denominator.
• Simplify the numerator by combining like terms and factoring out perfect squares.
• Simplify the denominator by combining like terms and factoring out perfect squares.
• Use algebraic identities and formulas to simplify the expression.
• Check your work by plugging in values and testing the expression.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying expressions with square roots:

• Failing to rationalize the denominator
• Failing to simplify the numerator and denominator
• Making algebraic errors, such as multiplying or dividing incorrectly
• Failing to check your work
• Not using algebraic identities and formulas to simplify the expression.

Conclusion

In conclusion, simplifying expressions with square roots is an essential skill in mathematics. By understanding how to rationalize the denominator and simplify expressions with square roots, we can solve problems more efficiently and effectively. We can also develop critical thinking and problem-solving skills, which are essential for success in mathematics and other fields.

Introduction

In our previous article, we discussed how to simplify the expression $\frac{2 \sqrt{3}}{\sqrt{6}}$ to its simplest form. In this article, we will answer some common questions related to simplifying expressions with square roots.

Q1: What is the simplest form of $\frac{\sqrt{2}}{\sqrt{3}}$?

A1: To simplify this expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{3}$. This will eliminate the square root from the denominator.

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

Q2: How do I simplify $\frac{\sqrt{5}}{\sqrt{10}}$?

A2: To simplify this expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{10}$. This will eliminate the square root from the denominator.

$\frac{\sqrt{5}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{5} \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}} = \frac{\sqrt{50}}{10} = \frac{5 \sqrt{2}}{10} = \frac{\sqrt{2}}{2}$

Q3: What is the simplest form of $\frac{3 \sqrt{2}}{\sqrt{8}}$?

A3: To simplify this expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{8}$. This will eliminate the square root from the denominator.

$\frac{3 \sqrt{2}}{\sqrt{8}} \cdot \frac{\sqrt{8}}{\sqrt{8}} = \frac{3 \sqrt{2} \cdot \sqrt{8}}{\sqrt{8} \cdot \sqrt{8}} = \frac{3 \sqrt{16}}{8} = \frac{3 \cdot 4}{8} = \frac{12}{8} = \frac{3}{2}$

Q4: How do I simplify $\frac{\sqrt{7}}{\sqrt{14}}$?

A4: To simplify this expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{14}$. This will eliminate the square root from the denominator.

$\frac{\sqrt{7}}{\sqrt{14}} \cdot \frac{\sqrt{14}}{\sqrt{14}} = \frac{\sqrt{7} \cdot \sqrt{14}}{\sqrt{14} \cdot \sqrt{14}} = \frac{\sqrt{98}}{14} = \frac{7 \sqrt{2}}{14} = \frac{\sqrt{2}}{2}$

Q5: What is the simplest form of $\frac{2 \sqrt{3}}{\sqrt{9}}$?

A5: To simplify this expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by $\sqrt{9}$. This will eliminate the square root from the denominator.

$\frac{2 \sqrt{3}}{\sqrt{9}} \cdot \frac{\sqrt{9}}{\sqrt{9}} = \frac{2 \sqrt{3} \cdot \sqrt{9}}{\sqrt{9} \cdot \sqrt{9}} = \frac{2 \sqrt{27}}{9} = \frac{2 \cdot 3 \sqrt{3}}{9} = \frac{6 \sqrt{3}}{9} = \frac{2 \sqrt{3}}{3}$

Conclusion

In this article, we answered some common questions related to simplifying expressions with square roots. We provided step-by-step solutions to each question, using various mathematical techniques, including rationalizing the denominator and simplifying expressions with square roots. By understanding how to simplify expressions with square roots, we can solve problems more efficiently and effectively, and develop critical thinking and problem-solving skills.

Final Answer

The final answers are:

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