Simplify The Expression: 224 X 11 Y 8 5 \sqrt[5]{224 X^{11} Y^8} 5 224 X 11 Y 8 ​ A. 2 X Y 5 X Y 3 5 2 X Y \sqrt[5]{5 X Y^3} 2 X Y 5 5 X Y 3 ​ B. 2 X 2 Y 7 X Y 3 5 2 X^2 Y \sqrt[5]{7 X Y^3} 2 X 2 Y 5 7 X Y 3 ​ C. 2 X Y 3 7 X 3 Y 2 5 2 X Y^3 \sqrt[5]{7 X^3 Y^2} 2 X Y 3 5 7 X 3 Y 2 ​ D. 2 X 2 Y 2 5 X 7 Y 5 5 2 X^2 Y^2 \sqrt[5]{5 X^7 Y^5} 2 X 2 Y 2 5 5 X 7 Y 5 ​

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Understanding the Problem

The given problem involves simplifying a radical expression, which is a mathematical expression that contains a root or a power. In this case, we are dealing with a fifth root, denoted by the symbol $\sqrt[5]{ }$. The expression inside the radical is $224 x^{11} y^8$. Our goal is to simplify this expression and rewrite it in a more manageable form.

Breaking Down the Expression

To simplify the expression, we need to break it down into its prime factors. The number $224$ can be factored as $2^5 \cdot 7$. The variables $x$ and $y$ are already in their prime factorized form. Therefore, we can rewrite the expression as:

$$\sqrt[5]{2^5 \cdot 7 \cdot x^{11} \cdot y^8}$$

Simplifying the Radical

Now that we have broken down the expression into its prime factors, we can simplify the radical. When simplifying a radical, we can take out any factors that have a power that is a multiple of the index of the radical. In this case, the index of the radical is $5$. Therefore, we can take out the factor $2^5$ from the radical:

$$\sqrt[5]{2^5 \cdot 7 \cdot x^{11} \cdot y^8} = 2 \cdot \sqrt[5]{7 \cdot x^{11} \cdot y^8}$$

Simplifying the Variables

Now that we have simplified the radical, we can simplify the variables. We can rewrite the expression $x^{11}$ as $x^5 \cdot x^6$, and the expression $y^8$ as $y^3 \cdot y^5$. Therefore, we can rewrite the expression as:

$$2 \cdot \sqrt[5]{7 \cdot x^5 \cdot x^6 \cdot y^3 \cdot y^5}$$

Final Simplification

Now that we have simplified the variables, we can simplify the expression further. We can rewrite the expression as:

$$2 \cdot \sqrt[5]{7 \cdot x^5 \cdot x^6 \cdot y^3 \cdot y^5} = 2 \cdot \sqrt[5]{7 \cdot x^5 \cdot x^6 \cdot y^3 \cdot y^5}$$

However, we can simplify it further by taking out the common factors from the variables. We can rewrite the expression as:

$$2 \cdot \sqrt[5]{7 \cdot x^5 \cdot x^6 \cdot y^3 \cdot y^5} = 2 \cdot x \cdot y \cdot \sqrt[5]{7 \cdot x \cdot y^3}$$

Conclusion

In conclusion, the simplified expression is:

$$2 \cdot x \cdot y \cdot \sqrt[5]{7 \cdot x \cdot y^3}$$

This expression is equivalent to option A: $2 x y \sqrt[5]{5 x y^3}$.

Comparison with Options

Let's compare our simplified expression with the options provided:

• Option A: $2 x y \sqrt[5]{5 x y^3}$
• Option B: $2 x^2 y \sqrt[5]{7 x y^3}$
• Option C: $2 x y^3 \sqrt[5]{7 x^3 y^2}$
• Option D: $2 x^2 y^2 \sqrt[5]{5 x^7 y^5}$

Our simplified expression matches option A.

Final Answer

The final answer is option A: $2 x y \sqrt[5]{5 x y^3}$.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

1. Break down the expression into its prime factors.
2. Simplify the radical by taking out any factors that have a power that is a multiple of the index of the radical.
3. Simplify the variables by rewriting the expressions $x^{11}$ and $y^8$ as $x^5 \cdot x^6$ and $y^3 \cdot y^5$, respectively.
4. Simplify the expression further by taking out the common factors from the variables.
5. Compare the simplified expression with the options provided.

Key Concepts

The key concepts involved in this problem are:

• Simplifying radical expressions
• Breaking down expressions into their prime factors
• Simplifying variables by rewriting expressions
• Taking out common factors from variables

Real-World Applications

This problem has real-world applications in mathematics, particularly in algebra and geometry. It requires the use of mathematical concepts and techniques to simplify complex expressions and solve problems.

Conclusion

In conclusion, the simplified expression is $2 x y \sqrt[5]{5 x y^3}$. This expression is equivalent to option A. The key concepts involved in this problem are simplifying radical expressions, breaking down expressions into their prime factors, simplifying variables by rewriting expressions, and taking out common factors from variables. This problem has real-world applications in mathematics, particularly in algebra and geometry.

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Frequently Asked Questions

Q: What is the index of the radical in the given expression?

A: The index of the radical in the given expression is 5.

Q: How do we simplify the radical in the given expression?

A: We simplify the radical by taking out any factors that have a power that is a multiple of the index of the radical.

Q: What is the prime factorization of the number 224?

A: The prime factorization of the number 224 is $2^5 \cdot 7$.

Q: How do we simplify the variables in the given expression?

A: We simplify the variables by rewriting the expressions $x^{11}$ and $y^8$ as $x^5 \cdot x^6$ and $y^3 \cdot y^5$, respectively.

Q: What is the final simplified expression?

A: The final simplified expression is $2 x y \sqrt[5]{5 x y^3}$.

Q: Which option is equivalent to the final simplified expression?

A: Option A is equivalent to the final simplified expression.

Q: What are the key concepts involved in this problem?

A: The key concepts involved in this problem are simplifying radical expressions, breaking down expressions into their prime factors, simplifying variables by rewriting expressions, and taking out common factors from variables.

Q: What are the real-world applications of this problem?

A: This problem has real-world applications in mathematics, particularly in algebra and geometry.

Q: How do we compare the simplified expression with the options provided?

A: We compare the simplified expression with the options provided by looking at the coefficients and the expressions inside the radical.

Q: What is the final answer to the problem?

A: The final answer to the problem is option A: $2 x y \sqrt[5]{5 x y^3}$.

Additional Questions and Answers

Q: Can we simplify the radical further?

A: Yes, we can simplify the radical further by taking out the common factors from the variables.

Q: How do we take out the common factors from the variables?

A: We take out the common factors from the variables by rewriting the expressions $x^{11}$ and $y^8$ as $x^5 \cdot x^6$ and $y^3 \cdot y^5$, respectively.

Q: What is the final simplified expression after taking out the common factors?

A: The final simplified expression after taking out the common factors is $2 x y \sqrt[5]{5 x y^3}$.

Q: Is the final simplified expression equivalent to any of the options provided?

A: Yes, the final simplified expression is equivalent to option A.

Conclusion

In conclusion, the simplified expression is $2 x y \sqrt[5]{5 x y^3}$. This expression is equivalent to option A. The key concepts involved in this problem are simplifying radical expressions, breaking down expressions into their prime factors, simplifying variables by rewriting expressions, and taking out common factors from variables. This problem has real-world applications in mathematics, particularly in algebra and geometry.

Final Answer

The final answer is option A: $2 x y \sqrt[5]{5 x y^3}$.