Find The Simplified Product:${ \sqrt[3]{9x^4} \cdot \sqrt[3]{3x^8} }$A. { \sqrt[3]{12x^{12}}$}$B. { \sqrt[3]{27x^{12}}$}$C. ${ 3x^4\$} D. ${ 9x^6\$}

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Understanding Radical Expressions

Radical expressions are mathematical expressions that involve the use of roots or radicals. In this article, we will focus on simplifying radical expressions, specifically the product of two cube roots. We will explore the properties of radical expressions and provide a step-by-step guide on how to simplify them.

Properties of Radical Expressions

Radical expressions have several properties that can be used to simplify them. One of the most important properties is the product rule, which states that the product of two radical expressions is equal to the product of their radicands. In other words, if we have two radical expressions an\sqrt[n]{a} and bn\sqrt[n]{b}, then their product is equal to abn\sqrt[n]{ab}.

Simplifying the Product of Two Cube Roots

Now, let's apply the product rule to the given expression: 9x43â‹…3x83\sqrt[3]{9x^4} \cdot \sqrt[3]{3x^8}. To simplify this expression, we need to multiply the radicands together.

9x43â‹…3x83=9x4â‹…3x83\sqrt[3]{9x^4} \cdot \sqrt[3]{3x^8} = \sqrt[3]{9x^4 \cdot 3x^8}

Multiplying the Radicands

To multiply the radicands, we need to multiply the coefficients and the variables separately.

9x4â‹…3x8=(9â‹…3)x4+89x^4 \cdot 3x^8 = (9 \cdot 3)x^{4+8} =27x12= 27x^{12}

Simplifying the Expression

Now that we have multiplied the radicands, we can simplify the expression by writing it in its simplest form.

9x43â‹…3x83=27x123\sqrt[3]{9x^4} \cdot \sqrt[3]{3x^8} = \sqrt[3]{27x^{12}}

Conclusion

In this article, we have learned how to simplify radical expressions using the product rule. We have applied this rule to the given expression and simplified it to its simplest form. The final answer is 27x123\sqrt[3]{27x^{12}}.

Comparison with Answer Choices

Now, let's compare our final answer with the answer choices provided.

A. 12x123\sqrt[3]{12x^{12}} B. 27x123\sqrt[3]{27x^{12}} C. 3x43x^4 D. 9x69x^6

Our final answer, 27x123\sqrt[3]{27x^{12}}, matches answer choice B.

Final Answer

The final answer is 27x123\boxed{\sqrt[3]{27x^{12}}}.

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about simplifying radical expressions. We will cover topics such as the product rule, multiplying radicands, and simplifying expressions.

Q: What is the product rule for radical expressions?

A: The product rule for radical expressions states that the product of two radical expressions is equal to the product of their radicands. In other words, if we have two radical expressions an\sqrt[n]{a} and bn\sqrt[n]{b}, then their product is equal to abn\sqrt[n]{ab}.

Q: How do I multiply the radicands in a radical expression?

A: To multiply the radicands in a radical expression, you need to multiply the coefficients and the variables separately. For example, if we have the expression 9x43â‹…3x83\sqrt[3]{9x^4} \cdot \sqrt[3]{3x^8}, we would multiply the coefficients (9 and 3) and the variables (x4x^4 and x8x^8) separately.

Q: What is the difference between a radicand and a coefficient?

A: A radicand is the number or expression inside the radical sign, while a coefficient is a number that is multiplied by the radicand. For example, in the expression 9x43\sqrt[3]{9x^4}, the radicand is 9x49x^4 and the coefficient is 9.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to follow these steps:

  1. Multiply the radicands together using the product rule.
  2. Simplify the resulting expression by combining like terms.
  3. Write the final answer in its simplest form.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is 27x123\sqrt[3]{27x^{12}}.

Q: Why is it important to simplify radical expressions?

A: Simplifying radical expressions is important because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to work with the expression
  • Avoid errors when working with the expression

Q: Can I use the product rule to simplify other types of radical expressions?

A: Yes, the product rule can be used to simplify other types of radical expressions, such as square roots and fourth roots.

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

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

  • Forgetting to multiply the coefficients and variables separately
  • Not simplifying the resulting expression
  • Writing the final answer in a way that is not in its simplest form

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through examples and exercises in a math textbook or online resource. You can also try creating your own examples and challenging yourself to simplify them.

Final Answer

The final answer is 27x123\boxed{\sqrt[3]{27x^{12}}}.