What Is The Following Quotient?A. $\frac{9+\sqrt{2}}{4-\sqrt{7}}$B. $\frac{9 \sqrt{7}+\sqrt{14}}{-3}$C. $\frac{36-9 \sqrt{7}+4 \sqrt{2}-\sqrt{14}}{9}$D. $\frac{36+9 \sqrt{7}+4 \sqrt{2}+\sqrt{14}}{9}$E.

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Mathematics is a vast and fascinating field that deals with numbers, quantities, and shapes. It is a fundamental subject that has numerous applications in various aspects of life, including science, technology, engineering, and mathematics (STEM). In this article, we will delve into the world of mathematics and explore the concept of quotients, specifically focusing on the given problem.

Understanding Quotients

A quotient is the result of dividing one number by another. It is a fundamental concept in mathematics that is used to find the ratio of two quantities. In the given problem, we are asked to find the quotient of a complex expression. To solve this problem, we need to apply various mathematical techniques, including algebraic manipulation and radical simplification.

Simplifying the Expression

The given expression is 9+24−7\frac{9+\sqrt{2}}{4-\sqrt{7}}. To simplify this expression, we need to rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of 4−74-\sqrt{7} is 4+74+\sqrt{7}.

\frac{9+\sqrt{2}}{4-\sqrt{7}} \cdot \frac{4+\sqrt{7}}{4+\sqrt{7}}

Multiplying the Numerator and the Denominator

To multiply the numerator and the denominator, we need to apply the distributive property. This involves multiplying each term in the numerator by each term in the denominator.

\frac{(9+\sqrt{2})(4+\sqrt{7})}{(4-\sqrt{7})(4+\sqrt{7})}

Expanding the Numerator and the Denominator

To expand the numerator and the denominator, we need to apply the distributive property. This involves multiplying each term in the numerator by each term in the denominator.

\frac{36+9\sqrt{7}+4\sqrt{2}+\sqrt{14}}{16-7}

Simplifying the Expression

To simplify the expression, we need to evaluate the denominator. The denominator is 16−716-7, which equals 99.

\frac{36+9\sqrt{7}+4\sqrt{2}+\sqrt{14}}{9}

Conclusion

In conclusion, the given quotient is 36+97+42+149\frac{36+9\sqrt{7}+4\sqrt{2}+\sqrt{14}}{9}. This expression cannot be simplified further, and it is the final answer to the problem.

Comparison with Other Options

To verify the correctness of our answer, we need to compare it with the other options. Let's analyze each option:

  • Option A: 9+24−7\frac{9+\sqrt{2}}{4-\sqrt{7}} is the original expression, and it is not equal to our answer.
  • Option B: 97+14−3\frac{9\sqrt{7}+\sqrt{14}}{-3} is not equal to our answer.
  • Option C: 36−97+42−149\frac{36-9\sqrt{7}+4\sqrt{2}-\sqrt{14}}{9} is not equal to our answer.
  • Option D: 36+97+42+149\frac{36+9\sqrt{7}+4\sqrt{2}+\sqrt{14}}{9} is equal to our answer.

Final Answer

Mathematics is a vast and fascinating field that deals with numbers, quantities, and shapes. It is a fundamental subject that has numerous applications in various aspects of life, including science, technology, engineering, and mathematics (STEM). In our previous article, we explored the concept of quotients, specifically focusing on the given problem. In this article, we will delve deeper into the world of mathematics and provide a Q&A guide to help you better understand the concept of quotients.

Q: What is a quotient?

A: A quotient is the result of dividing one number by another. It is a fundamental concept in mathematics that is used to find the ratio of two quantities.

Q: How do I simplify a quotient?

A: To simplify a quotient, you need to rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator.

Q: What is the conjugate of a denominator?

A: The conjugate of a denominator is the same expression with the opposite sign. For example, the conjugate of 4−74-\sqrt{7} is 4+74+\sqrt{7}.

Q: How do I multiply the numerator and the denominator?

A: To multiply the numerator and the denominator, you need to apply the distributive property. This involves multiplying each term in the numerator by each term in the denominator.

Q: What is the distributive property?

A: The distributive property is a mathematical concept that states that a single term can be distributed to multiple terms. For example, a(b+c)=ab+aca(b+c) = ab + ac.

Q: How do I expand the numerator and the denominator?

A: To expand the numerator and the denominator, you need to apply the distributive property. This involves multiplying each term in the numerator by each term in the denominator.

Q: What is the final answer to the problem?

A: The final answer to the problem is 36+97+42+149\boxed{\frac{36+9\sqrt{7}+4\sqrt{2}+\sqrt{14}}{9}}.

Common Mistakes to Avoid

When working with quotients, there are several common mistakes to avoid:

  • Not rationalizing the denominator: Failing to rationalize the denominator can lead to incorrect answers.
  • Not applying the distributive property: Failing to apply the distributive property can lead to incorrect answers.
  • Not simplifying the expression: Failing to simplify the expression can lead to incorrect answers.

Tips and Tricks

When working with quotients, here are some tips and tricks to keep in mind:

  • Use the conjugate to rationalize the denominator: The conjugate is a powerful tool for rationalizing the denominator.
  • Apply the distributive property: The distributive property is a fundamental concept in mathematics that can help you simplify expressions.
  • Simplify the expression: Simplifying the expression can help you find the final answer.

Conclusion

In conclusion, the concept of quotients is a fundamental concept in mathematics that has numerous applications in various aspects of life. By understanding the concept of quotients and applying the techniques outlined in this article, you can simplify complex expressions and find the final answer. Remember to rationalize the denominator, apply the distributive property, and simplify the expression to avoid common mistakes and find the correct answer.