Given X \textgreater 0 X \ \textgreater \ 0 X \textgreater 0 And Y \textgreater 0 Y \ \textgreater \ 0 Y \textgreater 0 , Select The Expression That Is Equivalent To 16 X 8 Y 10 4 \sqrt[4]{16 X^8 Y^{10}} 4 16 X 8 Y 10 ​ .Answer:A. 4 X 4 Y 6 4 X^4 Y^6 4 X 4 Y 6 B. 2 X 2 Y 5 2 2 X^2 Y^{\frac{5}{2}} 2 X 2 Y 2 5 ​ C. $4 X^2

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Introduction

Radical expressions, also known as roots, are an essential part of mathematics, particularly in algebra and geometry. In this article, we will focus on simplifying radical expressions, specifically the expression 16x8y104\sqrt[4]{16 x^8 y^{10}}. We will explore the properties of radical expressions, identify the equivalent expression, and provide a step-by-step guide on how to simplify it.

Understanding Radical Expressions

A radical expression is a mathematical expression that involves a root, such as a square root, cube root, or fourth root. The general form of a radical expression is an\sqrt[n]{a}, where nn is the index of the root and aa is the radicand. In this case, we have the expression 16x8y104\sqrt[4]{16 x^8 y^{10}}, where the index of the root is 4 and the radicand is 16x8y1016 x^8 y^{10}.

Properties of Radical Expressions

To simplify radical expressions, we need to understand the properties of radical expressions. The following are some key properties:

  • Product Property: an⋅bn=abn\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}
  • Quotient Property: anbn=abn\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}
  • Power Property: amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}
  • Zero Property: 0n=0\sqrt[n]{0} = 0
  • Negative Property: −an=±an\sqrt[n]{-a} = \pm \sqrt[n]{a}

Simplifying the Expression

Now that we have a good understanding of the properties of radical expressions, let's simplify the expression 16x8y104\sqrt[4]{16 x^8 y^{10}}. We can start by breaking down the radicand into its prime factors.

Breaking Down the Radicand

The radicand 16x8y1016 x^8 y^{10} can be broken down into its prime factors as follows:

16x8y10=24⋅x8⋅y1016 x^8 y^{10} = 2^4 \cdot x^8 \cdot y^{10}

Applying the Power Property

Now that we have broken down the radicand into its prime factors, we can apply the power property to simplify the expression.

16x8y104=24⋅x8⋅y104\sqrt[4]{16 x^8 y^{10}} = \sqrt[4]{2^4 \cdot x^8 \cdot y^{10}}

Using the power property, we can rewrite the expression as:

24⋅x8⋅y104=244⋅x84⋅y104\sqrt[4]{2^4 \cdot x^8 \cdot y^{10}} = 2^{\frac{4}{4}} \cdot x^{\frac{8}{4}} \cdot y^{\frac{10}{4}}

Simplifying further, we get:

244⋅x84⋅y104=21⋅x2⋅y522^{\frac{4}{4}} \cdot x^{\frac{8}{4}} \cdot y^{\frac{10}{4}} = 2^1 \cdot x^2 \cdot y^{\frac{5}{2}}

Conclusion

In conclusion, the expression 16x8y104\sqrt[4]{16 x^8 y^{10}} is equivalent to 2x2y522 x^2 y^{\frac{5}{2}}. We used the properties of radical expressions, specifically the power property, to simplify the expression. By breaking down the radicand into its prime factors and applying the power property, we were able to simplify the expression and find its equivalent form.

Answer

The correct answer is:

B. 2x2y522 x^2 y^{\frac{5}{2}}

Discussion

This problem requires a good understanding of the properties of radical expressions, specifically the power property. By breaking down the radicand into its prime factors and applying the power property, we were able to simplify the expression and find its equivalent form. This problem is a great example of how to simplify radical expressions and is a useful tool for anyone studying mathematics.

Additional Resources

For more information on radical expressions and how to simplify them, check out the following resources:

  • Khan Academy: Radical Expressions
  • Mathway: Simplifying Radical Expressions
  • Wolfram Alpha: Radical Expressions

Final Thoughts

Introduction

In our previous article, we explored the properties of radical expressions and how to simplify them. We used the power property to simplify the expression 16x8y104\sqrt[4]{16 x^8 y^{10}} and found its equivalent form. In this article, we will provide a Q&A guide to help you better understand radical expressions and how to simplify them.

Q: What is a radical expression?

A: A radical expression is a mathematical expression that involves a root, such as a square root, cube root, or fourth root. The general form of a radical expression is an\sqrt[n]{a}, where nn is the index of the root and aa is the radicand.

Q: What are the properties of radical expressions?

A: The properties of radical expressions include:

  • Product Property: an⋅bn=abn\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}
  • Quotient Property: anbn=abn\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}
  • Power Property: amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}
  • Zero Property: 0n=0\sqrt[n]{0} = 0
  • Negative Property: −an=±an\sqrt[n]{-a} = \pm \sqrt[n]{a}

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you can use the following steps:

  1. Break down the radicand into its prime factors.
  2. Apply the power property to simplify the expression.
  3. Simplify the resulting expression.

Q: What is the power property of radical expressions?

A: The power property of radical expressions states that amn=amn\sqrt[n]{a^m} = a^{\frac{m}{n}}. This means that you can simplify a radical expression by raising the radicand to the power of the index of the root.

Q: How do I apply the power property to simplify a radical expression?

A: To apply the power property, you can follow these steps:

  1. Break down the radicand into its prime factors.
  2. Raise each factor to the power of the index of the root.
  3. Simplify the resulting expression.

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

A: A radical expression is a mathematical expression that involves a root, such as a square root, cube root, or fourth root. A rational expression is a mathematical expression that involves a fraction, such as ab\frac{a}{b}. While both types of expressions involve variables and constants, they are distinct and require different techniques for simplification.

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

A: To simplify a radical expression with a variable in the radicand, you can use the following steps:

  1. Break down the radicand into its prime factors.
  2. Apply the power property to simplify the expression.
  3. Simplify the resulting expression.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

  • Not breaking down the radicand into its prime factors.
  • Not applying the power property correctly.
  • Not simplifying the resulting expression.

Conclusion

In conclusion, simplifying radical expressions is an essential part of mathematics, particularly in algebra and geometry. By understanding the properties of radical expressions and how to apply them, you can simplify complex expressions and find their equivalent forms. We hope this Q&A guide has been helpful in providing you with a better understanding of radical expressions and how to simplify them.

Additional Resources

For more information on radical expressions and how to simplify them, check out the following resources:

  • Khan Academy: Radical Expressions
  • Mathway: Simplifying Radical Expressions
  • Wolfram Alpha: Radical Expressions

Final Thoughts

Simplifying radical expressions is an essential part of mathematics, particularly in algebra and geometry. By understanding the properties of radical expressions and how to apply them, you can simplify complex expressions and find their equivalent forms. We hope this Q&A guide has been helpful in providing you with a better understanding of radical expressions and how to simplify them.