Which Choice Is Equivalent To The Product Below When $x \ \textgreater \ 0$?$\sqrt{\frac{2x}{6}} \cdot \sqrt{\frac{x}{3}}$A. $\frac{\sqrt{2x}}{9}$B. $\frac{x}{9}$C. $\frac{\sqrt{x}}{3}$D.

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Introduction

Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In this article, we will explore the process of simplifying radical expressions, with a focus on the given product 2x6â‹…x3\sqrt{\frac{2x}{6}} \cdot \sqrt{\frac{x}{3}}. We will examine each option and determine which one is equivalent to the given product when x>0x > 0.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or a higher root of a number. The square root of a number aa is denoted by a\sqrt{a} and is the number that, when multiplied by itself, gives the original number. For example, 16=4\sqrt{16} = 4 because 4×4=164 \times 4 = 16.

Simplifying the Given Product

To simplify the given product, we need to multiply the two square roots together. When multiplying square roots, we can combine the numbers inside the square roots by multiplying them together.

2x6â‹…x3=2x6â‹…x3\sqrt{\frac{2x}{6}} \cdot \sqrt{\frac{x}{3}} = \sqrt{\frac{2x}{6} \cdot \frac{x}{3}}

Now, we can simplify the fraction inside the square root by multiplying the numerators and denominators together.

2x6â‹…x3=2x218\sqrt{\frac{2x}{6} \cdot \frac{x}{3}} = \sqrt{\frac{2x^2}{18}}

Next, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2.

2x218=x29\sqrt{\frac{2x^2}{18}} = \sqrt{\frac{x^2}{9}}

Now, we can take the square root of the fraction by dividing the numerator and denominator by their greatest common divisor, which is xx.

x29=x3\sqrt{\frac{x^2}{9}} = \frac{x}{3}

Evaluating the Options

Now that we have simplified the given product, we can evaluate each option to determine which one is equivalent.

Option A: 2x9\frac{\sqrt{2x}}{9}

This option is not equivalent to the simplified product because the square root of 2x2x is not equal to xx.

Option B: x9\frac{x}{9}

This option is equivalent to the simplified product because it is equal to x3\frac{x}{3}.

Option C: x3\frac{\sqrt{x}}{3}

This option is not equivalent to the simplified product because the square root of xx is not equal to xx.

Option D: None of the above\boxed{\text{None of the above}}

This option is not equivalent to the simplified product because it is not equal to x3\frac{x}{3}.

Conclusion

In conclusion, the correct answer is B. x9\frac{x}{9}. This option is equivalent to the simplified product because it is equal to x3\frac{x}{3}. We hope this article has provided a clear and concise explanation of how to simplify radical expressions and evaluate options.

Additional Tips and Tricks

  • When simplifying radical expressions, always look for common factors that can be canceled out.
  • When multiplying square roots, combine the numbers inside the square roots by multiplying them together.
  • When simplifying fractions, divide both the numerator and denominator by their greatest common divisor.

Frequently Asked Questions

  • Q: What is the difference between a radical expression and a rational expression? A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while a rational expression is a mathematical expression that contains a fraction.
  • Q: How do I simplify a radical expression? A: To simplify a radical expression, look for common factors that can be canceled out, and then multiply the numbers inside the square roots together.
  • Q: What is the greatest common divisor of two numbers? A: The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder.
    Radical Expressions Q&A: Frequently Asked Questions and Answers ====================================================================

Introduction

Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In this article, we will provide a comprehensive Q&A section to help you better understand radical expressions and how to simplify them.

Q&A Section

Q: What is a radical expression?

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number.

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

A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while a rational expression is a mathematical expression that contains a fraction.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, look for common factors that can be canceled out, and then multiply the numbers inside the square roots together.

Q: What is the greatest common divisor of two numbers?

A: The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder.

Q: How do I multiply radical expressions?

A: To multiply radical expressions, combine the numbers inside the square roots by multiplying them together.

Q: How do I divide radical expressions?

A: To divide radical expressions, divide the numbers inside the square roots by multiplying the reciprocal of the divisor.

Q: What is the rule for multiplying radical expressions with different indices?

A: When multiplying radical expressions with different indices, multiply the numbers inside the square roots together and then multiply the indices together.

Q: What is the rule for dividing radical expressions with different indices?

A: When dividing radical expressions with different indices, divide the numbers inside the square roots by multiplying the reciprocal of the divisor and then divide the indices together.

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

A: To simplify a radical expression with a variable, look for common factors that can be canceled out and then multiply the numbers inside the square roots together.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the square of an integer, while a non-perfect square is a number that cannot be expressed as the square of an integer.

Q: How do I determine if a number is a perfect square?

A: To determine if a number is a perfect square, look for a whole number that, when multiplied by itself, gives the original number.

Q: What is the rule for simplifying radical expressions with perfect squares?

A: When simplifying radical expressions with perfect squares, cancel out the perfect square and then simplify the remaining expression.

Additional Tips and Tricks

  • When simplifying radical expressions, always look for common factors that can be canceled out.
  • When multiplying square roots, combine the numbers inside the square roots by multiplying them together.
  • When simplifying fractions, divide both the numerator and denominator by their greatest common divisor.

Frequently Asked Questions (FAQs)

  • Q: What is the difference between a radical expression and a rational expression? A: A radical expression is a mathematical expression that contains a square root or a higher root of a number, while a rational expression is a mathematical expression that contains a fraction.
  • Q: How do I simplify a radical expression? A: To simplify a radical expression, look for common factors that can be canceled out, and then multiply the numbers inside the square roots together.
  • Q: What is the greatest common divisor of two numbers? A: The greatest common divisor of two numbers is the largest number that divides both numbers without leaving a remainder.

Conclusion

In conclusion, we hope this Q&A article has provided a comprehensive guide to radical expressions and how to simplify them. Remember to always look for common factors that can be canceled out, and then multiply the numbers inside the square roots together. If you have any further questions or need additional help, don't hesitate to ask.