Select The Correct Answer.Simplify The Expression:$3x \sqrt[3]{648x^4y^8}$A. $18xy^2 \sqrt[3]{3x^2y^2}$ B. $18x^2y^2 \sqrt[3]{2xy^2}$ C. $18x^2y^2 \sqrt[3]{3xy^2}$ D. $9x^2y \sqrt[3]{2xy^2}$

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Understanding the Problem

To simplify the given expression, we need to apply the properties of radicals and exponents. The expression involves a cube root and a product of variables with exponents. Our goal is to rewrite the expression in a simpler form by combining like terms and applying the rules of radicals.

Breaking Down the Expression

The given expression is $3x \sqrt[3]{648x^4y^8}$. Let's break it down into smaller parts to simplify it step by step.

Step 1: Factor the Radicand

The radicand is $648x^4y^8$. We can factor $648$ as $2^3 \cdot 3^4$. So, the radicand can be written as:

$$648x^4y^8 = 2^3 \cdot 3^4 \cdot x^4 \cdot y^8$$

Step 2: Apply the Property of Radicals

We can rewrite the expression using the property of radicals that states $\sqrt[n]{a^n} = a$. Applying this property, we get:

$$\sqrt[3]{648x^4y^8} = \sqrt[3]{2^3 \cdot 3^4 \cdot x^4 \cdot y^8} = 2 \cdot 3 \cdot x \cdot y^2 \cdot \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$$

Step 3: Simplify the Expression

Now, we can simplify the expression by combining like terms:

$$3x \sqrt[3]{648x^4y^8} = 3x \cdot 2 \cdot 3 \cdot x \cdot y^2 \cdot \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$$

$$= 18x^2y^2 \cdot \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$$

Comparing with the Options

Now, let's compare the simplified expression with the given options:

A. $18xy^2 \sqrt[3]{3x^2y^2}$

B. $18x^2y^2 \sqrt[3]{2xy^2}$

C. $18x^2y^2 \sqrt[3]{3xy^2}$

D. $9x^2y \sqrt[3]{2xy^2}$

Conclusion

The simplified expression $18x^2y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$ matches option C. $18x^2y^2 \sqrt[3]{3xy^2}$.

The final answer is C.

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Understanding the Problem

To simplify the given expression, we need to apply the properties of radicals and exponents. The expression involves a cube root and a product of variables with exponents. Our goal is to rewrite the expression in a simpler form by combining like terms and applying the rules of radicals.

Q&A

Q: What is the radicand in the given expression?

A: The radicand is $648x^4y^8$.

Q: Can we factor the radicand?

A: Yes, we can factor $648$ as $2^3 \cdot 3^4$. So, the radicand can be written as:

$$648x^4y^8 = 2^3 \cdot 3^4 \cdot x^4 \cdot y^8$$

Q: How do we apply the property of radicals?

A: We can rewrite the expression using the property of radicals that states $\sqrt[n]{a^n} = a$. Applying this property, we get:

$$\sqrt[3]{648x^4y^8} = \sqrt[3]{2^3 \cdot 3^4 \cdot x^4 \cdot y^8} = 2 \cdot 3 \cdot x \cdot y^2 \cdot \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$$

Q: How do we simplify the expression?

A: Now, we can simplify the expression by combining like terms:

$$3x \sqrt[3]{648x^4y^8} = 3x \cdot 2 \cdot 3 \cdot x \cdot y^2 \cdot \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$$

$$= 18x^2y^2 \cdot \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$$

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

A: Now, let's compare the simplified expression with the given options:

A. $18xy^2 \sqrt[3]{3x^2y^2}$

B. $18x^2y^2 \sqrt[3]{2xy^2}$

C. $18x^2y^2 \sqrt[3]{3xy^2}$

D. $9x^2y \sqrt[3]{2xy^2}$

Q: Which option matches the simplified expression?

A: The simplified expression $18x^2y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$ matches option C. $18x^2y^2 \sqrt[3]{3xy^2}$.

Conclusion

The simplified expression $18x^2y^2 \sqrt[3]{3^2 \cdot x^2 \cdot y^2}$ matches option C. $18x^2y^2 \sqrt[3]{3xy^2}$.

Frequently Asked Questions

Q: What is the property of radicals that we used to simplify the expression?

A: The property of radicals that we used is $\sqrt[n]{a^n} = a$.

Q: How do we factor the radicand?

A: We can factor the radicand by breaking it down into its prime factors.

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

A: We compare the simplified expression with the given options by looking at the coefficients and the radicands.

Final Answer

The final answer is C.