Simplify The Expression: 15 4 − 3 8 \frac{\frac{\sqrt{15}}{4}}{-\frac{3}{8}} − 8 3 ​ 4 15 ​ ​ ​

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Introduction

When dealing with complex fractions, it can be challenging to simplify them, especially when they involve square roots and negative numbers. In this article, we will focus on simplifying the expression 154−38\frac{\frac{\sqrt{15}}{4}}{-\frac{3}{8}}. We will break down the problem into manageable steps, using a combination of mathematical concepts and techniques to arrive at the final solution.

Understanding the Problem

The given expression is a complex fraction, which means it is a fraction that contains another fraction in its numerator or denominator. In this case, the expression has a fraction in the numerator and a fraction in the denominator. Our goal is to simplify this expression by combining the fractions and eliminating any unnecessary steps.

Step 1: Simplify the Numerator

To simplify the expression, we start by simplifying the numerator. The numerator is 154\frac{\sqrt{15}}{4}. Since the square root of 15 cannot be simplified further, we leave it as is. However, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 4 and 1 is 1, so we cannot simplify the fraction further.

Step 2: Simplify the Denominator

Next, we simplify the denominator, which is −38-\frac{3}{8}. The negative sign in the denominator can be ignored for now, as it will be handled later. The fraction −38-\frac{3}{8} can be simplified by dividing the numerator and denominator by their GCD, which is 1. Therefore, the denominator remains the same.

Step 3: Combine the Fractions

Now that we have simplified the numerator and denominator, we can combine the fractions. To do this, we multiply the numerator by the reciprocal of the denominator. The reciprocal of −38-\frac{3}{8} is −83-\frac{8}{3}. Therefore, we multiply the numerator by −83-\frac{8}{3}.

Step 4: Simplify the Result

After multiplying the numerator by the reciprocal of the denominator, we get 154⋅−83=−15⋅84⋅3\frac{\sqrt{15}}{4} \cdot -\frac{8}{3} = -\frac{\sqrt{15} \cdot 8}{4 \cdot 3}. We can simplify this expression further by canceling out any common factors in the numerator and denominator. In this case, the numerator and denominator have no common factors, so the expression remains the same.

Step 5: Handle the Negative Sign

Finally, we need to handle the negative sign in the denominator. When we multiplied the numerator by the reciprocal of the denominator, we introduced a negative sign. To eliminate this negative sign, we can multiply both the numerator and denominator by −1-1. This will cancel out the negative sign in the denominator.

Step 6: Final Simplification

After handling the negative sign, we get 15⋅84⋅3=81512\frac{\sqrt{15} \cdot 8}{4 \cdot 3} = \frac{8\sqrt{15}}{12}. We can simplify this expression further by canceling out any common factors in the numerator and denominator. In this case, the numerator and denominator have a common factor of 4, so we can cancel it out.

Step 7: Final Answer

After canceling out the common factor, we get 2153\frac{2\sqrt{15}}{3}. This is the final simplified expression.

Conclusion

Simplifying complex fractions requires a combination of mathematical concepts and techniques. By breaking down the problem into manageable steps and using a combination of fraction simplification and square root manipulation, we can arrive at the final solution. In this article, we simplified the expression 154−38\frac{\frac{\sqrt{15}}{4}}{-\frac{3}{8}} and arrived at the final answer of 2153\frac{2\sqrt{15}}{3}.

Introduction

In our previous article, we simplified the expression 154−38\frac{\frac{\sqrt{15}}{4}}{-\frac{3}{8}} and arrived at the final answer of 2153\frac{2\sqrt{15}}{3}. However, we understand that simplifying complex fractions can be a challenging task, and many readers may have questions about the process. In this article, we will address some of the most frequently asked questions about simplifying complex fractions.

Q&A

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains another fraction in its numerator or denominator. In other words, it is a fraction that has a fraction within it.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, you need to follow these steps:

  1. Simplify the numerator and denominator separately.
  2. Combine the fractions by multiplying the numerator by the reciprocal of the denominator.
  3. Simplify the result by canceling out any common factors in the numerator and denominator.
  4. Handle any negative signs in the denominator.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and denominator. For example, the reciprocal of 34\frac{3}{4} is 43\frac{4}{3}.

Q: How do I handle negative signs in the denominator?

A: When you multiply the numerator by the reciprocal of the denominator, you may introduce a negative sign. To eliminate this negative sign, you can multiply both the numerator and denominator by −1-1.

Q: Can I simplify a complex fraction by canceling out common factors in the numerator and denominator?

A: Yes, you can simplify a complex fraction by canceling out common factors in the numerator and denominator. However, you need to be careful not to cancel out any factors that are not common to both the numerator and denominator.

Q: What is the difference between a complex fraction and a nested fraction?

A: A complex fraction is a fraction that contains another fraction in its numerator or denominator. A nested fraction, on the other hand, is a fraction that contains another fraction within it, but the inner fraction is not in the numerator or denominator.

Q: How do I simplify a nested fraction?

A: To simplify a nested fraction, you need to follow these steps:

  1. Simplify the inner fraction.
  2. Simplify the outer fraction.
  3. Combine the fractions by multiplying the numerator by the reciprocal of the denominator.
  4. Simplify the result by canceling out any common factors in the numerator and denominator.

Q: Can I use a calculator to simplify a complex fraction?

A: Yes, you can use a calculator to simplify a complex fraction. However, it is always a good idea to check your work by hand to ensure that the calculator is giving you the correct answer.

Conclusion

Simplifying complex fractions can be a challenging task, but with practice and patience, you can master the techniques. By following the steps outlined in this article, you can simplify even the most complex fractions and arrive at the final answer. Remember to always check your work by hand to ensure that the calculator is giving you the correct answer.

Additional Resources

If you are struggling to simplify complex fractions, there are many online resources available to help you. Some of these resources include:

  • Khan Academy: Khan Academy has a comprehensive video series on simplifying complex fractions.
  • Mathway: Mathway is an online calculator that can help you simplify complex fractions.
  • Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you simplify complex fractions.

Final Tips

  • Always simplify the numerator and denominator separately before combining the fractions.
  • Use the reciprocal of the denominator to combine the fractions.
  • Handle any negative signs in the denominator by multiplying both the numerator and denominator by −1-1.
  • Check your work by hand to ensure that the calculator is giving you the correct answer.

By following these tips and practicing regularly, you can become proficient in simplifying complex fractions and arrive at the final answer with confidence.