Which Choice Is Equivalent To The Product Below? 6 8 ⋅ 6 18 \sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}} 8 6 ​ ​ ⋅ 18 6 ​ ​ A. 6 12 \frac{6}{12} 12 6 ​ B. 1 4 \frac{1}{4} 4 1 ​ C. 7 12 \frac{7}{12} 12 7 ​ D. 3 4 \frac{3}{4} 4 3 ​ E. 7 1 \frac{7}{1} 1 7 ​

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the product of two square roots. We will examine the given expression $\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}}$ and determine which of the provided choices is equivalent to the product.

Understanding Square Roots

Before we dive into the simplification process, let's review the concept of square roots. A 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 $4 \cdot 4 = 16$. The square root of a fraction can be simplified by taking the square root of the numerator and the denominator separately.

Simplifying the Given Expression

Now, let's simplify the given expression $\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}}$. To do this, we will first simplify each square root individually.

Simplifying the First Square Root

The first square root is $\sqrt{\frac{6}{8}}$. We can simplify this by taking the square root of the numerator and the denominator separately:

$$\sqrt{\frac{6}{8}} = \frac{\sqrt{6}}{\sqrt{8}}$$

We can further simplify the denominator by recognizing that $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$:

$$\frac{\sqrt{6}}{\sqrt{8}} = \frac{\sqrt{6}}{2\sqrt{2}}$$

Simplifying the Second Square Root

The second square root is $\sqrt{\frac{6}{18}}$. We can simplify this by taking the square root of the numerator and the denominator separately:

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

We can further simplify the denominator by recognizing that $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$:

$$\frac{\sqrt{6}}{\sqrt{18}} = \frac{\sqrt{6}}{3\sqrt{2}}$$

Multiplying the Simplified Square Roots

Now that we have simplified each square root, we can multiply them together:

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

We can simplify the numerator by recognizing that $\sqrt{6} \cdot \sqrt{6} = 6$:

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

We can further simplify the denominator by recognizing that $2 \cdot 3 \cdot 2 = 12$:

$$\frac{6}{2 \cdot 3 \cdot 2} = \frac{6}{12}$$

Conclusion

In conclusion, the product of the two square roots $\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}}$ is equivalent to $\frac{6}{12}$. This is the correct answer among the provided choices.

Final Answer

The final answer is $\boxed{\frac{6}{12}}$.

Why is this the Correct Answer?

This is the correct answer because we simplified the given expression by taking the square root of the numerator and the denominator separately, and then multiplying the simplified square roots together. We recognized that $\sqrt{6} \cdot \sqrt{6} = 6$ and that $2 \cdot 3 \cdot 2 = 12$, which allowed us to simplify the expression to $\frac{6}{12}$.

What is the Importance of Simplifying Radical Expressions?

Simplifying radical expressions is an important skill in mathematics because it allows us to work with complex expressions in a more manageable way. By simplifying radical expressions, we can:

• Make calculations easier and more efficient
• Identify equivalent expressions and simplify them
• Solve equations and inequalities involving radical expressions
• Understand the properties of radical expressions and how they behave

Common Mistakes to Avoid

When simplifying radical expressions, there are several common mistakes to avoid:

• Not simplifying the numerator and denominator separately
• Not recognizing that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
• Not simplifying the expression by canceling out common factors
• Not checking the final answer to ensure it is equivalent to the original expression

Tips and Tricks

Here are some tips and tricks to help you simplify radical expressions:

• Always simplify the numerator and denominator separately
• Recognize that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
• Simplify the expression by canceling out common factors
• Check the final answer to ensure it is equivalent to the original expression
• Use a calculator or software to check your work and ensure accuracy

Conclusion

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Introduction

In our previous article, we explored the process of simplifying radical expressions, with a focus on the product of two square roots. We simplified the expression $\sqrt{\frac{6}{8}} \cdot \sqrt{\frac{6}{18}}$ and determined that the product is equivalent to $\frac{6}{12}$. In this article, we will answer some common questions about simplifying radical expressions and provide additional tips and tricks to help you master this important skill.

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

A: A radical expression is an expression that contains a square root or other root, while a rational expression is an expression that contains a fraction. For example, $\sqrt{6}$ is a radical expression, while $\frac{6}{12}$ is a rational expression.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

1. Simplify the numerator and denominator separately
2. Recognize that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
3. Simplify the expression by canceling out common factors
4. Check the final answer to ensure it is equivalent to the original expression

Q: What is the rule for multiplying radical expressions?

A: The rule for multiplying radical expressions is $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. This means that you can multiply the numbers inside the square roots together.

Q: How do I simplify a radical expression with a variable?

A: To simplify a radical expression with a variable, you need to follow the same steps as before:

1. Simplify the numerator and denominator separately
2. Recognize that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
3. Simplify the expression by canceling out common factors
4. Check the final answer to ensure it is equivalent to the original expression

Q: What is the difference between a rationalizing denominator and a rationalizing numerator?

A: A rationalizing denominator is a process of multiplying the numerator and denominator by a radical expression to eliminate the radical from the denominator. A rationalizing numerator is a process of multiplying the numerator and denominator by a radical expression to eliminate the radical from the numerator.

Q: How do I rationalize a denominator?

A: To rationalize a denominator, you need to follow these steps:

1. Multiply the numerator and denominator by a radical expression that will eliminate the radical from the denominator
2. Simplify the expression by canceling out common factors
3. Check the final answer to ensure it is equivalent to the original expression

Q: What is the difference between a conjugate and a rationalizing denominator?

A: A conjugate is a pair of binomials that have the same terms but opposite signs. A rationalizing denominator is a process of multiplying the numerator and denominator by a radical expression to eliminate the radical from the denominator.

Q: How do I find the conjugate of a binomial?

A: To find the conjugate of a binomial, you need to follow these steps:

1. Identify the binomial
2. Change the sign of the second term
3. Write the resulting binomial as the conjugate

Q: What is the importance of simplifying radical expressions?

A: Simplifying radical expressions is an important skill in mathematics because it allows you to work with complex expressions in a more manageable way. By simplifying radical expressions, you can:

• Make calculations easier and more efficient
• Identify equivalent expressions and simplify them
• Solve equations and inequalities involving radical expressions
• Understand the properties of radical expressions and how they behave

Conclusion

In conclusion, simplifying radical expressions is an important skill in mathematics that allows you to work with complex expressions in a more manageable way. By following the steps outlined in this article, you can simplify radical expressions and identify equivalent expressions. Remember to always simplify the numerator and denominator separately, recognize that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, and simplify the expression by canceling out common factors. With practice and patience, you can become proficient in simplifying radical expressions and solving equations and inequalities involving radical expressions.

Final Tips and Tricks

Here are some final tips and tricks to help you simplify radical expressions:

• Always simplify the numerator and denominator separately
• Recognize that $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$
• Simplify the expression by canceling out common factors
• Check the final answer to ensure it is equivalent to the original expression
• Use a calculator or software to check your work and ensure accuracy
• Practice, practice, practice! The more you practice simplifying radical expressions, the more comfortable you will become with the process.