Perform The Indicated Operations And Write The Result In Standard Form.${ \frac{-20+\sqrt{-50}}{60} }$ { \frac{-20+\sqrt{-50}}{60} = \square \} (Type An Exact Answer, Using Radicals As Needed. Type Your Answer In The Form [$ A

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Introduction

In mathematics, rationalizing the denominator is a process used to eliminate any radicals from the denominator of a fraction. This is particularly important when dealing with complex fractions, as it allows us to simplify the expression and make it easier to work with. In this article, we will explore the process of rationalizing the denominator and apply it to the given expression: −20+−5060\frac{-20+\sqrt{-50}}{60}.

Understanding the Problem

The given expression is a fraction with a radical in the numerator and a whole number in the denominator. Our goal is to simplify this expression by rationalizing the denominator, which means removing any radicals from the denominator. To do this, we need to multiply the numerator and denominator by a clever choice of expression that will eliminate the radical in the denominator.

Step 1: Simplify the Radical in the Numerator

Before we can rationalize the denominator, we need to simplify the radical in the numerator. The radical in the numerator is −50\sqrt{-50}. To simplify this radical, we can factor out the largest perfect square from the expression under the radical sign.

−50=−1⋅50=−1⋅50=i50\sqrt{-50} = \sqrt{-1 \cdot 50} = \sqrt{-1} \cdot \sqrt{50} = i\sqrt{50}

where ii is the imaginary unit, which is defined as the square root of -1.

Step 2: Rationalize the Denominator

Now that we have simplified the radical in the numerator, we can rationalize the denominator. To do this, we need to multiply the numerator and denominator by a clever choice of expression that will eliminate the radical in the denominator. In this case, we can multiply the numerator and denominator by 60\sqrt{60}.

−20+i5060⋅6060=−2060+i50606060\frac{-20+i\sqrt{50}}{60} \cdot \frac{\sqrt{60}}{\sqrt{60}} = \frac{-20\sqrt{60}+i\sqrt{50}\sqrt{60}}{60\sqrt{60}}

Step 3: Simplify the Expression

Now that we have rationalized the denominator, we can simplify the expression. To do this, we can combine like terms in the numerator and simplify the denominator.

−2060+i50606060=−204⋅15+i504⋅15604⋅15\frac{-20\sqrt{60}+i\sqrt{50}\sqrt{60}}{60\sqrt{60}} = \frac{-20\sqrt{4 \cdot 15}+i\sqrt{50}\sqrt{4 \cdot 15}}{60\sqrt{4 \cdot 15}}

=−20⋅215+i50⋅21560⋅215= \frac{-20 \cdot 2\sqrt{15}+i\sqrt{50} \cdot 2\sqrt{15}}{60 \cdot 2\sqrt{15}}

=−4015+2i501512015= \frac{-40\sqrt{15}+2i\sqrt{50}\sqrt{15}}{120\sqrt{15}}

=−4015+2i501512015= \frac{-40\sqrt{15}+2i\sqrt{50}\sqrt{15}}{120\sqrt{15}}

=−4015+2i250012015= \frac{-40\sqrt{15}+2i\sqrt{2500}}{120\sqrt{15}}

=−4015+2i50212015= \frac{-40\sqrt{15}+2i\sqrt{50^2}}{120\sqrt{15}}

=−4015+2i⋅5012015= \frac{-40\sqrt{15}+2i \cdot 50}{120\sqrt{15}}

=−4015+100i12015= \frac{-40\sqrt{15}+100i}{120\sqrt{15}}

Conclusion

In this article, we have explored the process of rationalizing the denominator and applied it to the given expression: −20+−5060\frac{-20+\sqrt{-50}}{60}. We simplified the radical in the numerator, rationalized the denominator, and simplified the expression. The final result is −4015+100i12015\frac{-40\sqrt{15}+100i}{120\sqrt{15}}. This expression is in standard form, with no radicals in the denominator.

Final Answer

Introduction

In our previous article, we explored the process of rationalizing the denominator and applied it to the given expression: −20+−5060\frac{-20+\sqrt{-50}}{60}. In this article, we will answer some frequently asked questions about rationalizing the denominator and provide additional examples to help you understand the concept.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is a process used to eliminate any radicals from the denominator of a fraction. This is particularly important when dealing with complex fractions, as it allows us to simplify the expression and make it easier to work with.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it allows us to simplify complex fractions and make them easier to work with. It also helps us to avoid dealing with fractions that have radicals in the denominator, which can be difficult to work with.

Q: How do I rationalize the denominator of a fraction?

A: To rationalize the denominator of a fraction, you need to multiply the numerator and denominator by a clever choice of expression that will eliminate the radical in the denominator. This expression is usually a radical that is the conjugate of the denominator.

Q: What is the conjugate of a denominator?

A: The conjugate of a denominator is an expression that is the same as the denominator, but with the opposite sign. For example, if the denominator is a+ba + b, the conjugate is a−ba - b.

Q: How do I find the conjugate of a denominator?

A: To find the conjugate of a denominator, you need to change the sign of the second term. For example, if the denominator is a+ba + b, the conjugate is a−ba - b.

Q: Can I rationalize the denominator of a fraction with a variable in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a variable in the denominator. However, you need to be careful when multiplying the numerator and denominator by the conjugate, as this can lead to a more complex expression.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

  • Not multiplying the numerator and denominator by the conjugate
  • Not simplifying the expression after rationalizing the denominator
  • Not checking for any remaining radicals in the denominator

Q: Can I use a calculator to rationalize the denominator of a fraction?

A: Yes, you can use a calculator to rationalize the denominator of a fraction. However, it's always a good idea to check your work by hand to make sure that the expression is simplified correctly.

Q: How do I know if I have rationalized the denominator correctly?

A: To know if you have rationalized the denominator correctly, you need to check that there are no radicals in the denominator. You can do this by simplifying the expression and checking that the denominator is a whole number or a simple radical.

Conclusion

In this article, we have answered some frequently asked questions about rationalizing the denominator and provided additional examples to help you understand the concept. We hope that this article has been helpful in clarifying any confusion you may have had about rationalizing the denominator.

Final Tips

  • Always simplify the expression after rationalizing the denominator
  • Check for any remaining radicals in the denominator
  • Use a calculator to check your work, but always double-check by hand

Common Examples

  • 1+22\frac{1+\sqrt{2}}{2}
  • 3−54\frac{3-\sqrt{5}}{4}
  • 2+35\frac{2+\sqrt{3}}{5}

Practice Problems

  • Rationalize the denominator of the following fractions:
    • 1+32\frac{1+\sqrt{3}}{2}
    • 2−53\frac{2-\sqrt{5}}{3}
    • 3+24\frac{3+\sqrt{2}}{4}

Answer Key

  • 1+32=1+32â‹…1−31−3=1−3+1−32−3=−23−1=23\frac{1+\sqrt{3}}{2} = \frac{1+\sqrt{3}}{2} \cdot \frac{1-\sqrt{3}}{1-\sqrt{3}} = \frac{1-\sqrt{3}+1-\sqrt{3}}{2-3} = \frac{-2\sqrt{3}}{-1} = 2\sqrt{3}
  • 2−53=2−53â‹…2+52+5=4+25−25−53(2+5)=−13(2+5)\frac{2-\sqrt{5}}{3} = \frac{2-\sqrt{5}}{3} \cdot \frac{2+\sqrt{5}}{2+\sqrt{5}} = \frac{4+2\sqrt{5}-2\sqrt{5}-5}{3(2+\sqrt{5})} = \frac{-1}{3(2+\sqrt{5})}
  • 3+24=3+24â‹…3−23−2=9−22+32−24(3−2)=7+24(3−2)\frac{3+\sqrt{2}}{4} = \frac{3+\sqrt{2}}{4} \cdot \frac{3-\sqrt{2}}{3-\sqrt{2}} = \frac{9-2\sqrt{2}+3\sqrt{2}-2}{4(3-\sqrt{2})} = \frac{7+\sqrt{2}}{4(3-\sqrt{2})}