What Is The Following Quotient?A. 1 1 + 3 \frac{1}{1+\sqrt{3}} 1 + 3 ​ 1 ​ B. 3 4 \frac{\sqrt{3}}{4} 4 3 ​ ​ C. 1 + 3 4 \frac{1+\sqrt{3}}{4} 4 1 + 3 ​ ​ D. 1 − 3 4 \frac{1-\sqrt{3}}{4} 4 1 − 3 ​ ​ E. − 1 + 3 2 \frac{-1+\sqrt{3}}{2} 2 − 1 + 3 ​ ​

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Rationalizing the Denominator

To find the value of the given quotient, we need to rationalize the denominator. Rationalizing the denominator involves getting rid of any radicals in the denominator. In this case, we have a quotient with a denominator that contains a square root.

Step 1: Multiply the Numerator and Denominator by the Conjugate

To rationalize the denominator, we need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression a+ba + b is a−ba - b. In this case, the denominator is 1+31 + \sqrt{3}, so the conjugate is 1−31 - \sqrt{3}.

\frac{1}{1+\sqrt{3}} \cdot \frac{1-\sqrt{3}}{1-\sqrt{3}}

Step 2: Simplify the Expression

Now, we can simplify the expression by multiplying the numerators and denominators.

\frac{1(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})}

Step 3: Apply the Difference of Squares Formula

The denominator can be simplified using the difference of squares formula, which states that (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. In this case, we have (1+3)(1−3)=12−(3)2=1−3=−2(1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2.

\frac{1-\sqrt{3}}{-2}

Step 4: Simplify the Expression

Now, we can simplify the expression by dividing the numerator by the denominator.

\frac{1-\sqrt{3}}{-2} = \frac{-1+\sqrt{3}}{2}

Conclusion

Therefore, the value of the given quotient is −1+32\frac{-1+\sqrt{3}}{2}.

Comparison with Answer Choices

Let's compare our answer with the answer choices.

  • A. 11+3\frac{1}{1+\sqrt{3}} is not equal to our answer.
  • B. 34\frac{\sqrt{3}}{4} is not equal to our answer.
  • C. 1+34\frac{1+\sqrt{3}}{4} is not equal to our answer.
  • D. 1−34\frac{1-\sqrt{3}}{4} is not equal to our answer.
  • E. −1+32\frac{-1+\sqrt{3}}{2} is equal to our answer.

Therefore, the correct answer is E. −1+32\frac{-1+\sqrt{3}}{2}.

Final Answer

The final answer is −1+32\boxed{\frac{-1+\sqrt{3}}{2}}.

Frequently Asked Questions

Q: What is the quotient in question?

A: The quotient in question is 11+3\frac{1}{1+\sqrt{3}}.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a binomial expression a+ba + b is a−ba - b. In this case, the denominator is 1+31 + \sqrt{3}, so the conjugate is 1−31 - \sqrt{3}.

Q: What is the conjugate of the denominator?

A: The conjugate of the denominator 1+31 + \sqrt{3} is 1−31 - \sqrt{3}.

Q: How do I simplify the expression after multiplying by the conjugate?

A: After multiplying by the conjugate, you can simplify the expression by multiplying the numerators and denominators. Then, you can apply the difference of squares formula to simplify the denominator.

Q: What is the difference of squares formula?

A: The difference of squares formula states that (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. In this case, we have (1+3)(1−3)=12−(3)2=1−3=−2(1+\sqrt{3})(1-\sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2.

Q: How do I simplify the expression after applying the difference of squares formula?

A: After applying the difference of squares formula, you can simplify the expression by dividing the numerator by the denominator.

Q: What is the final answer?

A: The final answer is −1+32\frac{-1+\sqrt{3}}{2}.

Q: How do I compare my answer with the answer choices?

A: To compare your answer with the answer choices, you need to look at each answer choice and determine if it is equal to your answer. In this case, the correct answer is E. −1+32\frac{-1+\sqrt{3}}{2}.

Q: What if I get a different answer than the one provided?

A: If you get a different answer than the one provided, you should recheck your work to ensure that you made no mistakes. You can also try to simplify the expression in a different way to see if you get the same answer.

Q: Can I use a calculator to solve the problem?

A: While it is possible to use a calculator to solve the problem, it is not recommended. Calculators can be prone to errors, and it is often better to solve the problem by hand to ensure that you understand the steps involved.

Q: How do I know if my answer is correct?

A: To know if your answer is correct, you need to compare it with the answer provided. If your answer matches the answer provided, then you know that your answer is correct. If your answer does not match the answer provided, then you need to recheck your work to ensure that you made no mistakes.

Common Mistakes to Avoid

Mistake 1: Not Rationalizing the Denominator

One common mistake to avoid is not rationalizing the denominator. Rationalizing the denominator is an important step in solving the problem, and it is essential to get it right.

Mistake 2: Simplifying the Expression Incorrectly

Another common mistake to avoid is simplifying the expression incorrectly. When simplifying the expression, you need to make sure that you are following the correct steps and that you are not making any mistakes.

Mistake 3: Not Comparing the Answer with the Answer Choices

A third common mistake to avoid is not comparing the answer with the answer choices. It is essential to compare your answer with the answer choices to ensure that you are getting the correct answer.

Conclusion

In conclusion, solving the quotient in question requires careful attention to detail and a thorough understanding of the steps involved. By following the steps outlined in this article, you can ensure that you get the correct answer and avoid common mistakes. Remember to rationalize the denominator, simplify the expression correctly, and compare your answer with the answer choices to ensure that you are getting the correct answer.