Simplify: $\sqrt{50 X^3 Y}$A. $5|x| \sqrt{x Y}$ B. $5 X \sqrt{2 X Y}$ C. $5|x| \sqrt{2 X Y}$ D. $5 X \sqrt{2 Y}$

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Introduction

Radical expressions are a fundamental concept in algebra, and simplifying them is a crucial skill for any math enthusiast. In this article, we will focus on simplifying the radical expression 50x3y\sqrt{50 x^3 y}, and explore the different options available. We will break down the process into manageable steps, and provide a clear explanation of each step.

Understanding the Radical Expression

Before we dive into simplifying the radical expression, let's first understand what it means. The expression 50x3y\sqrt{50 x^3 y} represents the square root of the product of 50, x3x^3, and yy. In other words, it is the value that, when multiplied by itself, gives the product of 50, x3x^3, and yy.

Step 1: Factor the Number Under the Radical

The first step in simplifying the radical expression is to factor the number under the radical sign. In this case, we can factor 50 as 2×522 \times 5^2. This gives us:

50x3y=2×52×x3×y\sqrt{50 x^3 y} = \sqrt{2 \times 5^2 \times x^3 \times y}

Step 2: Simplify the Radical Expression

Now that we have factored the number under the radical sign, we can simplify the radical expression. We can take the square root of the perfect square factor, which is 525^2. This gives us:

2×52×x3×y=52×x3×y\sqrt{2 \times 5^2 \times x^3 \times y} = 5 \sqrt{2 \times x^3 \times y}

Step 3: Simplify the Expression Inside the Radical

Now that we have simplified the radical expression, we can simplify the expression inside the radical. We can factor out x2x^2 from the expression x3x^3, which gives us:

52×x3×y=52×x2×x×y5 \sqrt{2 \times x^3 \times y} = 5 \sqrt{2 \times x^2 \times x \times y}

Step 4: Simplify the Final Expression

Now that we have simplified the expression inside the radical, we can simplify the final expression. We can take the square root of the perfect square factor, which is x2x^2. This gives us:

52×x2×x×y=5x2×x×y5 \sqrt{2 \times x^2 \times x \times y} = 5 x \sqrt{2 \times x \times y}

Conclusion

In conclusion, simplifying the radical expression 50x3y\sqrt{50 x^3 y} involves factoring the number under the radical sign, simplifying the radical expression, simplifying the expression inside the radical, and simplifying the final expression. By following these steps, we can simplify the radical expression and arrive at the final answer.

Answer Options

Now that we have simplified the radical expression, let's compare our answer with the options provided:

  • A. 5∣x∣xy5|x| \sqrt{x y}
  • B. 5x2xy5 x \sqrt{2 x y}
  • C. 5∣x∣2xy5|x| \sqrt{2 x y}
  • D. 5x2y5 x \sqrt{2 y}

Our answer matches option B, which is 5x2xy5 x \sqrt{2 x y}.

Final Answer

The final answer is B\boxed{B}.

Additional Tips and Tricks

  • When simplifying radical expressions, always factor the number under the radical sign.
  • When simplifying the radical expression, always take the square root of the perfect square factor.
  • When simplifying the expression inside the radical, always factor out the perfect square factor.
  • When simplifying the final expression, always take the square root of the perfect square factor.

Introduction

In our previous article, we explored the process of simplifying radical expressions, focusing on the expression 50x3y\sqrt{50 x^3 y}. We broke down the process into manageable steps and arrived at the final answer. In this article, we will provide a Q&A guide to help you better understand the process of simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or a higher root. It is a way of representing a value that, when multiplied by itself, gives the product of the numbers inside the radical sign.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, follow these steps:

  1. Factor the number under the radical sign.
  2. Simplify the radical expression by taking the square root of the perfect square factor.
  3. Simplify the expression inside the radical by factoring out the perfect square factor.
  4. Simplify the final expression by taking the square root of the perfect square factor.

Q: What is a perfect square factor?

A: A perfect square factor is a factor that can be expressed as the square of an integer. For example, 222^2, 323^2, and 424^2 are all perfect square factors.

Q: How do I identify a perfect square factor?

A: To identify a perfect square factor, look for a factor that can be expressed as the square of an integer. For example, if you have the expression 16x2y\sqrt{16 x^2 y}, you can identify the perfect square factor as 424^2.

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

A: Yes, you can simplify a radical expression with a negative number under the radical sign. However, you must take the absolute value of the number and simplify the expression accordingly.

Q: Can I simplify a radical expression with a variable under the radical sign?

A: Yes, you can simplify a radical expression with a variable under the radical sign. However, you must follow the same steps as before, factoring the number under the radical sign, simplifying the radical expression, simplifying the expression inside the radical, and simplifying the final 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 factoring the number under the radical sign.
  • Not taking the square root of the perfect square factor.
  • Not factoring out the perfect square factor from the expression inside the radical.
  • Not simplifying the final expression.

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through examples and exercises. You can also use online resources, such as math websites and apps, to practice simplifying radical expressions.

Conclusion

In conclusion, simplifying radical expressions is a crucial skill for any math enthusiast. By following the steps outlined in this article, you can simplify radical expressions with ease and arrive at the final answer. Remember to factor the number under the radical sign, simplify the radical expression, simplify the expression inside the radical, and simplify the final expression. With practice and patience, you can become proficient in simplifying radical expressions.

Additional Resources

  • Math websites and apps, such as Khan Academy and Mathway, offer interactive exercises and tutorials to help you practice simplifying radical expressions.
  • Online communities, such as Reddit's r/learnmath and r/math, offer a platform to ask questions and receive feedback from other math enthusiasts.
  • Textbooks and workbooks, such as "Algebra and Trigonometry" by Michael Sullivan, offer comprehensive coverage of radical expressions and simplification techniques.