What Is The Following Quotient?${ \frac{\sqrt{96}}{\sqrt{8}} }$A. ${ 2 \sqrt{3}\$} B. 4 C. ${ 2 \sqrt{22}\$} D. 12

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Understanding the Quotient

When dealing with square roots in mathematics, it's essential to understand the properties and rules that govern their behavior. In this case, we're given a quotient involving square roots, and we need to simplify it to find the final answer.

Simplifying the Quotient

To simplify the quotient, we can start by simplifying the square roots individually. The square root of 96 can be simplified as follows:

96=16×6=16×6=46\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}

Similarly, the square root of 8 can be simplified as follows:

8=4×2=4×2=22\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}

Simplifying the Quotient

Now that we have simplified the square roots, we can substitute these values back into the original quotient:

968=4622\frac{\sqrt{96}}{\sqrt{8}} = \frac{4\sqrt{6}}{2\sqrt{2}}

Canceling Out Common Factors

We can simplify the quotient further by canceling out common factors. In this case, we can cancel out the 2 in the numerator and denominator:

4622=262\frac{4\sqrt{6}}{2\sqrt{2}} = \frac{2\sqrt{6}}{\sqrt{2}}

Rationalizing the Denominator

To rationalize the denominator, we can multiply both the numerator and denominator by the square root of 2:

262=26×22×2=2122\frac{2\sqrt{6}}{\sqrt{2}} = \frac{2\sqrt{6} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{12}}{2}

Simplifying the Square Root

The square root of 12 can be simplified as follows:

12=4×3=4×3=23\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}

Final Answer

Substituting this value back into the quotient, we get:

2122=2×232=23\frac{2\sqrt{12}}{2} = \frac{2 \times 2\sqrt{3}}{2} = 2\sqrt{3}

Conclusion

Therefore, the final answer to the given quotient is 232\sqrt{3}.

Comparison with Answer Choices

Let's compare our final answer with the answer choices provided:

A. 232\sqrt{3} B. 4 C. 2222\sqrt{22} D. 12

Our final answer matches answer choice A, which is 232\sqrt{3}.

Final Thoughts

In this problem, we used the properties of square roots to simplify the quotient. We started by simplifying the square roots individually, then canceled out common factors, and finally rationalized the denominator. By following these steps, we were able to simplify the quotient and find the final answer.

Key Takeaways

  • Simplify square roots individually before combining them.
  • Cancel out common factors to simplify the quotient.
  • Rationalize the denominator to eliminate any radicals in the denominator.

Common Mistakes to Avoid

  • Failing to simplify square roots individually before combining them.
  • Not canceling out common factors to simplify the quotient.
  • Not rationalizing the denominator to eliminate any radicals in the denominator.

Real-World Applications

Understanding how to simplify quotients involving square roots is essential in various real-world applications, such as:

  • Physics: When dealing with wave functions and energy levels.
  • Engineering: When designing and analyzing electrical circuits and mechanical systems.
  • Computer Science: When working with algorithms and data structures.

Conclusion

In conclusion, simplifying quotients involving square roots requires a thorough understanding of the properties and rules that govern their behavior. By following the steps outlined in this article, we can simplify even the most complex quotients and find the final answer.

Q: What is the first step in simplifying a quotient involving square roots?

A: The first step in simplifying a quotient involving square roots is to simplify the square roots individually. This involves breaking down the square root into its prime factors and simplifying it as much as possible.

Q: How do I simplify a square root of a number that is not a perfect square?

A: To simplify a square root of a number that is not a perfect square, you can break it down into its prime factors and simplify it as much as possible. For example, 96\sqrt{96} can be simplified as follows:

96=16×6=16×6=46\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}

Q: What is the difference between simplifying a square root and rationalizing the denominator?

A: Simplifying a square root involves breaking down the square root into its prime factors and simplifying it as much as possible. Rationalizing the denominator involves multiplying both the numerator and denominator by the square root of a number to eliminate any radicals in the denominator.

Q: How do I rationalize the denominator of a quotient involving square roots?

A: To rationalize the denominator of a quotient involving square roots, you can multiply both the numerator and denominator by the square root of a number that will eliminate any radicals in the denominator. For example, to rationalize the denominator of 262\frac{2\sqrt{6}}{\sqrt{2}}, you can multiply both the numerator and denominator by 2\sqrt{2}:

262=26×22×2=2122\frac{2\sqrt{6}}{\sqrt{2}} = \frac{2\sqrt{6} \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{12}}{2}

Q: What is the final step in simplifying a quotient involving square roots?

A: The final step in simplifying a quotient involving square roots is to simplify the resulting expression as much as possible. This may involve canceling out common factors, combining like terms, and simplifying any remaining radicals.

Q: Can I simplify a quotient involving square roots by canceling out common factors?

A: Yes, you can simplify a quotient involving square roots by canceling out common factors. For example, to simplify the quotient 4622\frac{4\sqrt{6}}{2\sqrt{2}}, you can cancel out the 2 in the numerator and denominator:

4622=262\frac{4\sqrt{6}}{2\sqrt{2}} = \frac{2\sqrt{6}}{\sqrt{2}}

Q: What are some common mistakes to avoid when simplifying quotients involving square roots?

A: Some common mistakes to avoid when simplifying quotients involving square roots include:

  • Failing to simplify square roots individually before combining them.
  • Not canceling out common factors to simplify the quotient.
  • Not rationalizing the denominator to eliminate any radicals in the denominator.

Q: How do I know if a quotient involving square roots can be simplified?

A: A quotient involving square roots can be simplified if it meets the following conditions:

  • The numerator and denominator can be simplified separately.
  • The square roots in the numerator and denominator can be simplified separately.
  • The resulting expression can be simplified further by canceling out common factors or rationalizing the denominator.

Q: Can I use a calculator to simplify a quotient involving square roots?

A: Yes, you can use a calculator to simplify a quotient involving square roots. However, it's always a good idea to check your work by simplifying the expression manually to ensure that you get the correct answer.

Q: How do I apply the concepts of simplifying quotients involving square roots to real-world problems?

A: The concepts of simplifying quotients involving square roots can be applied to a wide range of real-world problems, including:

  • Physics: When dealing with wave functions and energy levels.
  • Engineering: When designing and analyzing electrical circuits and mechanical systems.
  • Computer Science: When working with algorithms and data structures.

By understanding how to simplify quotients involving square roots, you can apply these concepts to a wide range of problems and make more accurate predictions and calculations.