Which Choice Is Equivalent To The Product Below When $x \ \textgreater \ 0$?$\sqrt{\frac{2}{x^2}} \cdot \sqrt{\frac{x^2}{18}}$A. $\frac{1}{9}$B. $\frac{x}{9}$C. $\frac{1}{3}$D. $\frac{x}{3}$

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Introduction

Radical expressions are a fundamental concept in 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 problem: 2x2β‹…x218\sqrt{\frac{2}{x^2}} \cdot \sqrt{\frac{x^2}{18}}. We will break down the solution step by step, using the properties of radicals and exponents to arrive at the final answer.

Understanding Radical Expressions

A radical expression is a mathematical expression that contains a square root or other root. 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.

Simplifying Radical Expressions

To simplify a radical expression, we need to use the properties of radicals and exponents. The two main properties we will use are:

  • Product of Radicals: aβ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}
  • Quotient of Radicals: ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}

Step 1: Simplify the First Radical Expression

The first radical expression is 2x2\sqrt{\frac{2}{x^2}}. We can simplify this expression by using the property of radicals: a=a12\sqrt{a} = a^{\frac{1}{2}}. Therefore, we can rewrite the expression as:

2x2=(2x2)12\sqrt{\frac{2}{x^2}} = \left(\frac{2}{x^2}\right)^{\frac{1}{2}}

Step 2: Simplify the Second Radical Expression

The second radical expression is x218\sqrt{\frac{x^2}{18}}. We can simplify this expression by using the property of radicals: a=a12\sqrt{a} = a^{\frac{1}{2}}. Therefore, we can rewrite the expression as:

x218=(x218)12\sqrt{\frac{x^2}{18}} = \left(\frac{x^2}{18}\right)^{\frac{1}{2}}

Step 3: Multiply the Two Simplified Radical Expressions

Now that we have simplified both radical expressions, we can multiply them together. Using the property of radicals: aβ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}, we can rewrite the expression as:

2x2β‹…x218=2x2β‹…x218\sqrt{\frac{2}{x^2}} \cdot \sqrt{\frac{x^2}{18}} = \sqrt{\frac{2}{x^2} \cdot \frac{x^2}{18}}

Step 4: Simplify the Expression Inside the Radical

The expression inside the radical is 2x2β‹…x218\frac{2}{x^2} \cdot \frac{x^2}{18}. We can simplify this expression by canceling out the x2x^2 terms:

2x2β‹…x218=218\frac{2}{x^2} \cdot \frac{x^2}{18} = \frac{2}{18}

Step 5: Simplify the Final Expression

The final expression is 218\sqrt{\frac{2}{18}}. We can simplify this expression by using the property of radicals: a=a12\sqrt{a} = a^{\frac{1}{2}}. Therefore, we can rewrite the expression as:

218=(218)12\sqrt{\frac{2}{18}} = \left(\frac{2}{18}\right)^{\frac{1}{2}}

Conclusion

In conclusion, the final answer to the problem is (218)12\left(\frac{2}{18}\right)^{\frac{1}{2}}. We can simplify this expression further by using the property of radicals: a=a12\sqrt{a} = a^{\frac{1}{2}}. Therefore, we can rewrite the expression as:

(218)12=218\left(\frac{2}{18}\right)^{\frac{1}{2}} = \frac{\sqrt{2}}{\sqrt{18}}

Simplifying the Final Expression

We can simplify the final expression by using the property of radicals: a=a12\sqrt{a} = a^{\frac{1}{2}}. Therefore, we can rewrite the expression as:

218=29β‹…2\frac{\sqrt{2}}{\sqrt{18}} = \frac{\sqrt{2}}{\sqrt{9} \cdot \sqrt{2}}

Simplifying the Final Expression (continued)

We can simplify the final expression by canceling out the 2\sqrt{2} terms:

29β‹…2=19\frac{\sqrt{2}}{\sqrt{9} \cdot \sqrt{2}} = \frac{1}{\sqrt{9}}

Simplifying the Final Expression (continued)

We can simplify the final expression by using the property of radicals: a=a12\sqrt{a} = a^{\frac{1}{2}}. Therefore, we can rewrite the expression as:

19=13\frac{1}{\sqrt{9}} = \frac{1}{3}

Final Answer

Q: What is the product of radicals property?

A: The product of radicals property states that aβ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.

Q: What is the quotient of radicals property?

A: The quotient of radicals property states that ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to use the properties of radicals and exponents. The two main properties you will use are the product of radicals and the quotient of radicals.

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

A: A radical expression is a mathematical expression that contains a square root or other root, while an exponential expression is a mathematical expression that contains a power or exponent.

Q: Can I simplify a radical expression by canceling out terms?

A: Yes, you can simplify a radical expression by canceling out terms. For example, if you have the expression 2x2β‹…x218\sqrt{\frac{2}{x^2}} \cdot \sqrt{\frac{x^2}{18}}, you can cancel out the x2x^2 terms to simplify the expression.

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

A: To simplify a radical expression with a fraction, you need to use the properties of radicals and fractions. You can simplify the fraction by canceling out common terms, and then simplify the radical expression using the product of radicals property.

Q: Can I simplify a radical expression with a negative number?

A: Yes, you can simplify a radical expression with a negative number. However, you need to be careful when simplifying the expression, as the negative sign can affect the final result.

Q: How do I know when to simplify a radical expression?

A: You should simplify a radical expression when it is necessary to do so. Simplifying a radical expression can make it easier to work with and can help you to arrive at the final answer more quickly.

Q: Can I use a calculator to simplify a radical expression?

A: Yes, you can use a calculator to simplify a radical expression. However, you need to be careful when using a calculator, as it may not always give you the correct result.

Q: How do I check my work when simplifying a radical expression?

A: To check your work when simplifying a radical expression, you need to make sure that you have used the correct properties of radicals and exponents. You can also check your work by plugging in values for the variables and seeing if the expression simplifies correctly.

Conclusion

In conclusion, simplifying radical expressions is an important skill to master in mathematics. By using the properties of radicals and exponents, you can simplify radical expressions and arrive at the final answer more quickly. Remember to be careful when simplifying radical expressions, as the negative sign and fractions can affect the final result.