What Is The Following Quotient?${ \frac{6-3(\sqrt[3]{6})}{\sqrt[3]{9}} }$A. ${ 2(\sqrt[3]{3})-\sqrt[3]{18}\$} B. ${ 2(\sqrt[3]{3})-3(\sqrt[3]{2})\$} C. ${ 3(\sqrt[3]{3})-\sqrt[3]{18}\$} D.
Understanding the Problem
The given problem involves evaluating a quotient that contains cube roots. To solve this, we need to carefully apply the rules of arithmetic operations, including the order of operations and the properties of cube roots.
Breaking Down the Quotient
The quotient we need to evaluate is:
{ \frac{6-3(\sqrt[3]{6})}{\sqrt[3]{9}} \}
To simplify this expression, we can start by evaluating the cube root terms inside the parentheses.
Evaluating the Cube Root Terms
The cube root of 6 can be written as , and the cube root of 9 can be written as . We can rewrite the quotient as:
{ \frac{6-3(\sqrt[3]{6})}{\sqrt[3]{9}} = \frac{6-3(\sqrt[3]{6})}{\sqrt[3]{9^2}} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{9}} \}
Simplifying the Quotient
Now, we can simplify the quotient by canceling out the common factors in the numerator and denominator.
{ \frac{6-3(\sqrt[3]{6})}{\sqrt[3]{9^2}} \cdot \frac{\sqrt[3]{9}}{\sqrt[3]{9}} = \frac{6-3(\sqrt[3]{6})}{\sqrt[3]{81}} \cdot \sqrt[3]{9} \}
Evaluating the Cube Root of 81
The cube root of 81 can be evaluated as .
Substituting the Value of the Cube Root of 81
Now, we can substitute the value of the cube root of 81 into the quotient.
{ \frac{6-3(\sqrt[3]{6})}{3\sqrt[3]{3}} \cdot \sqrt[3]{9} \}
Simplifying the Quotient Further
We can simplify the quotient further by multiplying the numerator and denominator by .
{ \frac{6-3(\sqrt[3]{6})}{3\sqrt[3]{3}} \cdot \sqrt[3]{9} = \frac{(6-3(\sqrt[3]{6}))\sqrt[3]{3}}{3\sqrt[3]{9}} \}
Evaluating the Numerator
The numerator can be evaluated as .
Substituting the Value of the Numerator
Now, we can substitute the value of the numerator into the quotient.
{ \frac{(6-3(\sqrt[3]{6}))\sqrt[3]{3}}{3\sqrt[3]{9}} \}
Simplifying the Quotient Further
We can simplify the quotient further by evaluating the expression inside the parentheses.
{ (6-3(\sqrt[3]{6})) = 6 - 3\sqrt[3]{6} \}
Substituting the Value of the Expression Inside the Parentheses
Now, we can substitute the value of the expression inside the parentheses into the quotient.
{ \frac{(6-3\sqrt[3]{6})\sqrt[3]{3}}{3\sqrt[3]{9}} \}
Simplifying the Quotient Further
We can simplify the quotient further by multiplying the numerator and denominator by .
{ \frac{(6-3\sqrt[3]{6})\sqrt[3]{3}}{3\sqrt[3]{9}} = \frac{(6-3\sqrt[3]{6})\sqrt[3]{27}}{3\sqrt[3]{81}} \}
Evaluating the Numerator
The numerator can be evaluated as .
Substituting the Value of the Numerator
Now, we can substitute the value of the numerator into the quotient.
{ \frac{(6-3\sqrt[3]{6})\sqrt[3]{27}}{3\sqrt[3]{81}} \}
Simplifying the Quotient Further
We can simplify the quotient further by evaluating the expression inside the parentheses.
{ (6-3\sqrt[3]{6}) = 2\sqrt[3]{3} - 3\sqrt[3]{2} \}
Substituting the Value of the Expression Inside the Parentheses
Now, we can substitute the value of the expression inside the parentheses into the quotient.
{ \frac{(2\sqrt[3]{3} - 3\sqrt[3]{2})\sqrt[3]{27}}{3\sqrt[3]{81}} \}
Simplifying the Quotient Further
We can simplify the quotient further by evaluating the cube root of 27 and 81.
{ \sqrt[3]{27} = 3, \sqrt[3]{81} = 3 \}
Substituting the Value of the Cube Root of 27 and 81
Now, we can substitute the value of the cube root of 27 and 81 into the quotient.
{ \frac{(2\sqrt[3]{3} - 3\sqrt[3]{2})\cdot 3}{3\cdot 3} \}
Simplifying the Quotient Further
We can simplify the quotient further by canceling out the common factors in the numerator and denominator.
{ \frac{(2\sqrt[3]{3} - 3\sqrt[3]{2})\cdot 3}{3\cdot 3} = \frac{2\sqrt[3]{3} - 3\sqrt[3]{2}}{3} \}
Evaluating the Quotient
The quotient can be evaluated as .
Conclusion
The final answer to the given problem is .
Discussion
The given problem involves evaluating a quotient that contains cube roots. To solve this, we need to carefully apply the rules of arithmetic operations, including the order of operations and the properties of cube roots. The solution involves simplifying the quotient by canceling out common factors, evaluating the cube root terms, and applying the properties of arithmetic operations. The final answer is .
Q&A: Evaluating the Quotient
Q: What is the given quotient?
A: The given quotient is ${ \frac{6-3(\sqrt[3]{6})}{\sqrt[3]{9}} }$
Q: How do I simplify the quotient?
A: To simplify the quotient, we need to carefully apply the rules of arithmetic operations, including the order of operations and the properties of cube roots.
Q: What are the steps to simplify the quotient?
A: The steps to simplify the quotient are:
- Evaluate the cube root terms inside the parentheses.
- Simplify the quotient by canceling out common factors.
- Evaluate the cube root terms.
- Apply the properties of arithmetic operations.
Q: What is the final answer to the given problem?
A: The final answer to the given problem is .
Q: Why is it important to carefully apply the rules of arithmetic operations?
A: It is essential to carefully apply the rules of arithmetic operations to ensure that the quotient is simplified correctly.
Q: What are some common mistakes to avoid when simplifying the quotient?
A: Some common mistakes to avoid when simplifying the quotient include:
- Not evaluating the cube root terms correctly.
- Not canceling out common factors.
- Not applying the properties of arithmetic operations correctly.
Q: How can I practice simplifying quotients with cube roots?
A: You can practice simplifying quotients with cube roots by working through example problems and exercises.
Q: What are some real-world applications of simplifying quotients with cube roots?
A: Simplifying quotients with cube roots has real-world applications in fields such as engineering, physics, and mathematics.
Q: Can I use a calculator to simplify the quotient?
A: While a calculator can be used to simplify the quotient, it is essential to understand the underlying mathematical concepts and rules of arithmetic operations.
Q: How can I check my work when simplifying the quotient?
A: You can check your work by:
- Verifying that the quotient is simplified correctly.
- Checking that the final answer matches the expected answer.
- Using a calculator to verify the answer.
Q: What are some common errors to look out for when simplifying the quotient?
A: Some common errors to look out for when simplifying the quotient include:
- Not evaluating the cube root terms correctly.
- Not canceling out common factors.
- Not applying the properties of arithmetic operations correctly.
Q: How can I improve my skills in simplifying quotients with cube roots?
A: You can improve your skills in simplifying quotients with cube roots by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some additional resources for learning about simplifying quotients with cube roots?
A: Some additional resources for learning about simplifying quotients with cube roots include:
- Online tutorials and videos.
- Math textbooks and workbooks.
- Online communities and forums.
Q: Can I use technology to simplify the quotient?
A: Yes, you can use technology such as calculators and computer software to simplify the quotient.
Q: How can I use technology to simplify the quotient?
A: You can use technology to simplify the quotient by:
- Using a calculator to evaluate the cube root terms.
- Using computer software to simplify the quotient.
- Using online tools and resources to check your work.
Q: What are some benefits of using technology to simplify the quotient?
A: Some benefits of using technology to simplify the quotient include:
- Increased accuracy.
- Faster results.
- Improved understanding of mathematical concepts.
Q: What are some limitations of using technology to simplify the quotient?
A: Some limitations of using technology to simplify the quotient include:
- Dependence on technology.
- Limited understanding of mathematical concepts.
- Potential for errors.
Q: How can I balance the use of technology with mathematical understanding?
A: You can balance the use of technology with mathematical understanding by:
- Using technology to check your work.
- Understanding the underlying mathematical concepts.
- Practicing regularly to develop your skills.
Q: What are some additional tips for simplifying quotients with cube roots?
A: Some additional tips for simplifying quotients with cube roots include:
- Breaking down the problem into smaller steps.
- Checking your work regularly.
- Seeking help from a teacher or tutor.
Q: How can I apply the skills I learn in simplifying quotients with cube roots to real-world problems?
A: You can apply the skills you learn in simplifying quotients with cube roots to real-world problems by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some real-world applications of simplifying quotients with cube roots?
A: Simplifying quotients with cube roots has real-world applications in fields such as engineering, physics, and mathematics.
Q: How can I use the skills I learn in simplifying quotients with cube roots to improve my problem-solving skills?
A: You can use the skills you learn in simplifying quotients with cube roots to improve your problem-solving skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some additional resources for learning about problem-solving skills?
A: Some additional resources for learning about problem-solving skills include:
- Online tutorials and videos.
- Math textbooks and workbooks.
- Online communities and forums.
Q: Can I use the skills I learn in simplifying quotients with cube roots to improve my critical thinking skills?
A: Yes, you can use the skills you learn in simplifying quotients with cube roots to improve your critical thinking skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: How can I apply the skills I learn in simplifying quotients with cube roots to improve my critical thinking skills?
A: You can apply the skills you learn in simplifying quotients with cube roots to improve your critical thinking skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some real-world applications of improving critical thinking skills?
A: Improving critical thinking skills has real-world applications in fields such as business, law, and medicine.
Q: How can I use the skills I learn in simplifying quotients with cube roots to improve my communication skills?
A: You can use the skills you learn in simplifying quotients with cube roots to improve your communication skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some additional resources for learning about communication skills?
A: Some additional resources for learning about communication skills include:
- Online tutorials and videos.
- Communication textbooks and workbooks.
- Online communities and forums.
Q: Can I use the skills I learn in simplifying quotients with cube roots to improve my teamwork skills?
A: Yes, you can use the skills you learn in simplifying quotients with cube roots to improve your teamwork skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: How can I apply the skills I learn in simplifying quotients with cube roots to improve my teamwork skills?
A: You can apply the skills you learn in simplifying quotients with cube roots to improve your teamwork skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some real-world applications of improving teamwork skills?
A: Improving teamwork skills has real-world applications in fields such as business, engineering, and medicine.
Q: How can I use the skills I learn in simplifying quotients with cube roots to improve my leadership skills?
A: You can use the skills you learn in simplifying quotients with cube roots to improve your leadership skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some additional resources for learning about leadership skills?
A: Some additional resources for learning about leadership skills include:
- Online tutorials and videos.
- Leadership textbooks and workbooks.
- Online communities and forums.
Q: Can I use the skills I learn in simplifying quotients with cube roots to improve my time management skills?
A: Yes, you can use the skills you learn in simplifying quotients with cube roots to improve your time management skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: How can I apply the skills I learn in simplifying quotients with cube roots to improve my time management skills?
A: You can apply the skills you learn in simplifying quotients with cube roots to improve your time management skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some real-world applications of improving time management skills?
A: Improving time management skills has real-world applications in fields such as business, engineering, and medicine.
Q: How can I use the skills I learn in simplifying quotients with cube roots to improve my organization skills?
A: You can use the skills you learn in simplifying quotients with cube roots to improve your organization skills by:
- Practicing regularly.
- Working through example problems and exercises.
- Seeking help from a teacher or tutor.
Q: What are some additional resources for learning about organization skills?
A: Some additional resources for learning about organization skills include:
- Online tutorials