Simplify − 27 A 6 B 9 3 \sqrt[3]{-27 A^6 B^9} 3 − 27 A 6 B 9 ​ .A. − 3 A 2 B 3 -3 A^2 B^3 − 3 A 2 B 3 B. − 3 A B 3 -3 A B^3 − 3 A B 3 C. 3 A 2 B 3 3 A^2 B^3 3 A 2 B 3 D. − 3 A 3 B 3 -3 A^3 B^3 − 3 A 3 B 3

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Understanding the Problem

When dealing with radicals, it's essential to understand the properties of exponents and roots. In this problem, we're given the expression $\sqrt[3]{-27 a^6 b^9}$, and we need to simplify it. To simplify a radical, we need to find the cube root of the expression inside the radical sign.

Breaking Down the Expression

Let's break down the expression $\sqrt[3]{-27 a^6 b^9}$ into its prime factors. We can start by factoring out the negative sign, which can be written as $-1$. Then, we can factor out the perfect cube $27$, which is equal to $3^3$. We can also factor out the variables $a^6$ and $b^9$.

Factoring Out Perfect Cubes

We can factor out perfect cubes from the expression $a^6 b^9$. Since $a^6$ can be written as $(a^2)^3$, we can factor out $a^2$ from $a^6$. Similarly, since $b^9$ can be written as $(b^3)^3$, we can factor out $b^3$ from $b^9$.

Simplifying the Expression

Now that we've factored out the perfect cubes, we can simplify the expression. We can rewrite the expression as $\sqrt[3]{-1 \cdot 3^3 \cdot (a^2)^3 \cdot (b^3)^3}$. Using the property of radicals that $\sqrt[n]{a^n} = a$, we can simplify the expression further.

Applying the Property of Radicals

Using the property of radicals that $\sqrt[n]{a^n} = a$, we can simplify the expression $\sqrt[3]{-1 \cdot 3^3 \cdot (a^2)^3 \cdot (b^3)^3}$ as $-1 \cdot 3 \cdot a^2 \cdot b^3$. This simplifies to $-3 a^2 b^3$.

Conclusion

In conclusion, the simplified form of the expression $\sqrt[3]{-27 a^6 b^9}$ is $-3 a^2 b^3$. This is the correct answer among the options provided.

Final Answer

The final answer is $\boxed{-3 a^2 b^3}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Factor out the negative sign from the expression $\sqrt[3]{-27 a^6 b^9}$.
2. Factor out the perfect cube $27$ from the expression $\sqrt[3]{-27 a^6 b^9}$.
3. Factor out the variables $a^6$ and $b^9$ from the expression $\sqrt[3]{-27 a^6 b^9}$.
4. Factor out perfect cubes from the expression $a^6 b^9$.
5. Simplify the expression using the property of radicals that $\sqrt[n]{a^n} = a$.
6. Apply the property of radicals to simplify the expression further.

Common Mistakes

Here are some common mistakes to avoid when simplifying radicals:

• Not factoring out perfect cubes from the expression.
• Not applying the property of radicals that $\sqrt[n]{a^n} = a$.
• Not simplifying the expression further after applying the property of radicals.

Tips and Tricks

Here are some tips and tricks to help you simplify radicals:

• Always factor out perfect cubes from the expression.
• Always apply the property of radicals that $\sqrt[n]{a^n} = a$.
• Always simplify the expression further after applying the property of radicals.

Real-World Applications

Simplifying radicals has many real-world applications in mathematics and science. For example, it's used in calculus to simplify expressions and solve equations. It's also used in physics to simplify expressions and solve problems involving motion and energy.

Practice Problems

Here are some practice problems to help you practice simplifying radicals:

• Simplify $\sqrt[3]{64 a^9 b^{12}}$.
• Simplify $\sqrt[3]{-125 a^6 b^9}$.
• Simplify $\sqrt[3]{216 a^3 b^6}$.

Conclusion

In conclusion, simplifying radicals is an essential skill in mathematics and science. By following the steps outlined in this article, you can simplify radicals and solve problems involving motion and energy. Remember to always factor out perfect cubes from the expression and apply the property of radicals that $\sqrt[n]{a^n} = a$. With practice and patience, you can become proficient in simplifying radicals and solving problems involving motion and energy.

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Frequently Asked Questions

Q: What is the simplified form of $\sqrt[3]{-27 a^6 b^9}$?

A: The simplified form of $\sqrt[3]{-27 a^6 b^9}$ is $-3 a^2 b^3$.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to factor out perfect cubes from the expression and apply the property of radicals that $\sqrt[n]{a^n} = a$.

Q: What is the property of radicals that $\sqrt[n]{a^n} = a$?

A: The property of radicals that $\sqrt[n]{a^n} = a$ states that the nth root of a number raised to the power of n is equal to the number itself.

Q: How do I factor out perfect cubes from a radical expression?

A: To factor out perfect cubes from a radical expression, you need to identify the perfect cube factors of the expression and factor them out.

Q: What is a perfect cube factor?

A: A perfect cube factor is a factor that can be expressed as a perfect cube, such as $3^3$ or $a^3$.

Q: How do I apply the property of radicals to simplify a radical expression?

A: To apply the property of radicals to simplify a radical expression, you need to identify the perfect cube factors of the expression and simplify them using the property of radicals.

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

A: Some common mistakes to avoid when simplifying radicals include not factoring out perfect cubes from the expression, not applying the property of radicals that $\sqrt[n]{a^n} = a$, and not simplifying the expression further after applying the property of radicals.

Q: What are some tips and tricks for simplifying radicals?

A: Some tips and tricks for simplifying radicals include always factoring out perfect cubes from the expression, always applying the property of radicals that $\sqrt[n]{a^n} = a$, and always simplifying the expression further after applying the property of radicals.

Q: How do I practice simplifying radicals?

A: You can practice simplifying radicals by working through practice problems, such as simplifying $\sqrt[3]{64 a^9 b^{12}}$ or $\sqrt[3]{-125 a^6 b^9}$.

Q: What are some real-world applications of simplifying radicals?

A: Simplifying radicals has many real-world applications in mathematics and science, such as simplifying expressions and solving equations in calculus and physics.

Additional Resources

• For more information on simplifying radicals, see the article "Simplify $\sqrt[3]{-27 a^6 b^9}$".
• For practice problems and exercises, see the article "Practice Problems: Simplifying Radicals".
• For real-world applications of simplifying radicals, see the article "Real-World Applications of Simplifying Radicals".

Conclusion

In conclusion, simplifying radicals is an essential skill in mathematics and science. By following the steps outlined in this article and practicing with real-world examples, you can become proficient in simplifying radicals and solving problems involving motion and energy. Remember to always factor out perfect cubes from the expression and apply the property of radicals that $\sqrt[n]{a^n} = a$. With practice and patience, you can become a master of simplifying radicals and solving problems involving motion and energy.

Final Answer

The final answer is $\boxed{-3 a^2 b^3}$.

Step-by-Step Solution

Here's a step-by-step solution to the problem:

1. Factor out the negative sign from the expression $\sqrt[3]{-27 a^6 b^9}$.
2. Factor out the perfect cube $27$ from the expression $\sqrt[3]{-27 a^6 b^9}$.
3. Factor out the variables $a^6$ and $b^9$ from the expression $\sqrt[3]{-27 a^6 b^9}$.
4. Factor out perfect cubes from the expression $a^6 b^9$.
5. Simplify the expression using the property of radicals that $\sqrt[n]{a^n} = a$.
6. Apply the property of radicals to simplify the expression further.

Common Mistakes

Here are some common mistakes to avoid when simplifying radicals:

• Not factoring out perfect cubes from the expression.
• Not applying the property of radicals that $\sqrt[n]{a^n} = a$.
• Not simplifying the expression further after applying the property of radicals.

Tips and Tricks

Here are some tips and tricks to help you simplify radicals:

• Always factor out perfect cubes from the expression.
• Always apply the property of radicals that $\sqrt[n]{a^n} = a$.
• Always simplify the expression further after applying the property of radicals.

Real-World Applications

Simplifying radicals has many real-world applications in mathematics and science. For example, it's used in calculus to simplify expressions and solve equations. It's also used in physics to simplify expressions and solve problems involving motion and energy.

Practice Problems

Here are some practice problems to help you practice simplifying radicals:

• Simplify $\sqrt[3]{64 a^9 b^{12}}$.
• Simplify $\sqrt[3]{-125 a^6 b^9}$.
• Simplify $\sqrt[3]{216 a^3 b^6}$.

Conclusion

In conclusion, simplifying radicals is an essential skill in mathematics and science. By following the steps outlined in this article and practicing with real-world examples, you can become proficient in simplifying radicals and solving problems involving motion and energy. Remember to always factor out perfect cubes from the expression and apply the property of radicals that $\sqrt[n]{a^n} = a$. With practice and patience, you can become a master of simplifying radicals and solving problems involving motion and energy.