November 2, 2024
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R2 – Roots – Simple

This post provides some simple examples of R2 roots. The calculations behind some of the examples are also shown.

Examples

Firstly, here are some examples of the roots of i^.

  • √(i^) = (√2/2^ + √2/2i^) or (√2/2v + √2/2iv)              
  • 3√(i^) = iv  or (√3/2^ + ½i^) etc.
  • 4√(i^) = (0.924^ + 0.383i^) etc.
  • 5√(i^) = i^  or  (0.951^ + 0.309i^) etc.
  • 6√(i^) = (0.966^ + 0.259i^) etc.
  • 7√(i^) = iv or (0.975^ + 0.223i^)etc.
  • 8√(i^) = (0.981^ + 0.195i^) etc.
  • 9√(i^) = i^ or (0.985^ + 0.174i^)etc.
  • 10√(i^) = (0.988^ + 0.156i^) etc.

Secondly, here are examples of the roots of iv.

  • √(iv) = (√2/2^ + √2/2iv) or (√2/2v + √2/2i^)                
  • 3√(iv) = i^ or (√3/2^ + ½iv) etc.
  • 4√(iv) = (0.924^ + 0.383iv) etc.                                                 
  • 5√(iv) = iv  or (0.951^ + 0.309iv) etc.      

Finally, here are some examples of non unitary roots.

  • √(9^) = 3^or 3v;                                                                              
  • √(9v) = 3i^or 3iv  
  • √(9i^) = 2.12^+ 2.12i^ or 2.12v + 2.12iv   
  • √(9iv) = 2.12^+ 2.12iv or 2.12v + 2.12i^                  

Sample Calculations

These calculations show how these simple R2 roots can be multiplied in order to get the original Opposite Value.

  • 3√(1^) = (½v  + √3/2i^) or (½v + √3/2iv):
    • v  + √3/2i^)3  = (½v  + √3/2i^)(¼^ +  √3/4iv + √3/4iv + ¾v)
    • = (½v  + √3/2i^)(½v  +  √3/2iv)
    • = (¼^  +  √3/4i^ + √3/4iv + ¾^)
    • = 1^
    • v  + √3/2iv)3  = (½v  + √3/2iv )(¼^ +  √3/4i^ + √3/4i^ + ¾v)
    • = (½v  + √3/2iv )(½v +  √3/2i^)
    •  =  (¼^ +  √3/4iv + √3/4i^ + ¾^)
    • = 1^

  • 3√(i^) = (√3/2^ + i^/2):
    •  ((√3/2^) + (½i^))3 = ((√3/2^) + (½i^))(( √3/2^) + (½i^))(( √3/2^) + (½i^))
    • = ((√3/2^) + (½i^))(( √3/2^)(√3/2^) + (√3/2^)(½i^) + (½i^)( √3/2^) + (½i^)( ½i^))
    • = ((√3/2^) + (½i^))( ¾^ + √¾i^ + √¾i^ +  ¼v)
    • = ((√3/2^) + (½i^))( ½^ + √3/2i^)
    • = (√3/4^ + ¾i^ + ¼i^ + √3/4v)
    • = i^

  • 4√(i^) = (0.924^ + 0.383i^):
    • (0.924^ + 0.383i^)4 =   (0.924^*0.924^  + 0.924^*0.383i^ +  0.383i^*0.924^ +  0.383i^* 0.383i^)2
    • =  (0.854^ + 0.354i^ + 0.354i^ + 0.147v)2
    • =  (0.707^ +0.708i^)2
    • = 0.5^ + 0.5i^ + 0.5i^ + 0.5v  
    • = i^

  • 5√(i^) = (0.951^ + 0.309i^):
    • (0.951^+0.309i^)5= (0.951^+0.309i^)*(0.951^*0.951^ + 0.951^*0.309i^ + 0.309i^*0.951^ + 0.309i^*0.309i^)2
    • = (0.951^ + 0.309i^)*(0.904^ + 0.294i^ + 0.294i^ + 0.095v)2
    • =  (0.951^ + 0.309i^)*(0.809^ +0.588i^)2
    • =  (0.951^ + 0.309i^)*(0.654^ + 0.475i^ + 0.475i^ + 0.345v)
    • =  (0.951^ + 0.309i^)*(0.309^ + 0. 95i^)
    • =  (0.294^ + 0.903i^ + 0.095i^ + 0.294v)
    • =   i^

Conclusion

Try these examples of R2 roots with our online calculator.

                                                                                 

Next: Roots of Expressions

Previous: Definition

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