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

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Understanding the Problem

When simplifying an expression with radicals, it's essential to identify the prime factorization of the radicand and then simplify the expression using the properties of radicals. In this problem, we're given the expression $\sqrt[5]{224 x^{11} y^8}$ and asked to simplify it.

Breaking Down the Radicand

To simplify the expression, we need to break down the radicand, which is $224 x^{11} y^8$. We can start by finding the prime factorization of 224.

Prime Factorization of 224

The prime factorization of 224 is $2^5 \cdot 7$. Therefore, we can rewrite the radicand as:

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

Simplifying the Expression

Now that we have the prime factorization of the radicand, we can simplify the expression using the properties of radicals. We can rewrite the expression as:

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

Simplifying the Radicand

We can further simplify the radicand by breaking down the powers of x and y.

$$\sqrt[5]{7 \cdot x^{11} \cdot y^8} = \sqrt[5]{7 \cdot x^{10} \cdot x \cdot y^8}$$

Applying the Power Rule

Using the power rule of radicals, we can rewrite the expression as:

$$\sqrt[5]{7 \cdot x^{10} \cdot x \cdot y^8} = \sqrt[5]{7 \cdot x^{10}} \cdot \sqrt[5]{x \cdot y^8}$$

Simplifying the Radicals

We can further simplify the radicals by breaking down the powers of x and y.

$$\sqrt[5]{7 \cdot x^{10}} = \sqrt[5]{7} \cdot \sqrt[5]{x^{10}} = \sqrt[5]{7} \cdot x^2$$

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

Combining the Simplified Radicals

Now that we have simplified the radicals, we can combine them to get the final simplified expression.

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

Final Simplification

Using the property of radicals that $\sqrt[n]{a^n} = a$, we can simplify the expression further.

$$2 \cdot \sqrt[5]{7} \cdot x^2 \cdot \sqrt[5]{x} \cdot y^2 = 2 \cdot \sqrt[5]{7} \cdot x^{2 + 1/5} \cdot y^2$$

$$2 \cdot \sqrt[5]{7} \cdot x^{2 + 1/5} \cdot y^2 = 2 \cdot \sqrt[5]{7} \cdot x^{12/5} \cdot y^2$$

$$2 \cdot \sqrt[5]{7} \cdot x^{12/5} \cdot y^2 = 2 \cdot \sqrt[5]{7} \cdot x^{2.4} \cdot y^2$$

However, we can simplify it further by expressing the exponent as a mixed number.

$$2 \cdot \sqrt[5]{7} \cdot x^{2.4} \cdot y^2 = 2 \cdot \sqrt[5]{7} \cdot x^{24/10} \cdot y^2$$

$$2 \cdot \sqrt[5]{7} \cdot x^{24/10} \cdot y^2 = 2 \cdot \sqrt[5]{7} \cdot x^{12/5} \cdot y^2$$

However, we can simplify it further by expressing the exponent as a mixed number.

$$2 \cdot \sqrt[5]{7} \cdot x^{12/5} \cdot y^2 = 2 \cdot \sqrt[5]{7} \cdot x^{2} \cdot x^{1/5} \cdot y^2$$

$$2 \cdot \sqrt[5]{7} \cdot x^{2} \cdot x^{1/5} \cdot y^2 = 2 \cdot \sqrt[5]{7} \cdot x^{2} \cdot \sqrt[5]{x} \cdot y^2$$

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

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

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

Conclusion

The final simplified expression is $2 \cdot \sqrt[5]{7} \cdot x \cdot y^2$. This is the correct answer.

Answer

The correct answer is D. $2 x y^3 \sqrt[5]{7 x^3 y^2}$

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Understanding the Problem

When simplifying an expression with radicals, it's essential to identify the prime factorization of the radicand and then simplify the expression using the properties of radicals. In this problem, we're given the expression $\sqrt[5]{224 x^{11} y^8}$ and asked to simplify it.

Q&A

Q: What is the prime factorization of 224?

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

Q: How do we simplify the expression $\sqrt[5]{2^5 \cdot 7 \cdot x^{11} \cdot y^8}$?

A: We can simplify the expression by breaking down the powers of x and y, and then applying the power rule of radicals.

Q: What is the power rule of radicals?

A: The power rule of radicals states that $\sqrt[n]{a^n} = a$.

Q: How do we apply the power rule of radicals to the expression $\sqrt[5]{7 \cdot x^{10} \cdot x \cdot y^8}$?

A: We can rewrite the expression as $\sqrt[5]{7 \cdot x^{10}} \cdot \sqrt[5]{x \cdot y^8}$, and then simplify each radical separately.

Q: How do we simplify the radical $\sqrt[5]{7 \cdot x^{10}}$?

A: We can simplify the radical by breaking down the powers of x and y, and then applying the power rule of radicals.

Q: What is the simplified form of the radical $\sqrt[5]{7 \cdot x^{10}}$?

A: The simplified form of the radical is $\sqrt[5]{7} \cdot x^2$.

Q: How do we simplify the radical $\sqrt[5]{x \cdot y^8}$?

A: We can simplify the radical by breaking down the powers of x and y, and then applying the power rule of radicals.

Q: What is the simplified form of the radical $\sqrt[5]{x \cdot y^8}$?

A: The simplified form of the radical is $\sqrt[5]{x} \cdot y^2$.

Q: How do we combine the simplified radicals to get the final simplified expression?

A: We can combine the simplified radicals by multiplying them together.

Q: What is the final simplified expression?

A: The final simplified expression is $2 \cdot \sqrt[5]{7} \cdot x^2 \cdot \sqrt[5]{x} \cdot y^2$.

Q: Can we simplify the expression further?

A: Yes, we can simplify the expression further by combining the terms with the same base.

Q: What is the final simplified expression?

A: The final simplified expression is $2 \cdot \sqrt[5]{7} \cdot x \cdot y^2$.

Conclusion

The final simplified expression is $2 \cdot \sqrt[5]{7} \cdot x \cdot y^2$. This is the correct answer.

Answer

The correct answer is D. $2 x y^3 \sqrt[5]{7 x^3 y^2}$