July 22, 2024
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R2 – Roots – Expressions

This post looks at the roots of expressions in R2.

Calculate the full root of an R2 expression as the root of the Counter times the root of the Unitary. This applies whether the Counter is known or is a variable.

Example 1

  • √(x??) = √x*√(??)
  • For example if x = 16 and ?? = 1^:
    • √(16^) = √16*√1^ = 4*1^ = 4^ or 4*1v = 4v
    • √(16v) = √16*√1v = 4*i^ = 4i^ or 4*iv = 4iv

Example 2

  • 3√(x??) = 3√x*3√(??)
  • For example if x = 27 and ?? = 1^:
    • 3√(27^) = 3√27*√1^ = 3*1^ = 3^
      • or 3*(0.5v + 0.866i^) = (1.5v + 2.6i^)
      • or 3*(0.5v + 0.866iv) = (1.5v + 2.6iv)
    • 3√(27v) = 3√27*√1v = 3*1v = 3v
      • or 3*(0.5^ + 0.866i^) = (1.5^ + 2.6i^)
      • or 3*(0.5^ + 0.866iv) = (1.5^ + 2.6iv)

Example 3

The multiplications below show that (x^  + xi^)2 = (xv  + xiv)2 = 2x2i^

  • (x^  + xi^)2 = x2^ + 2x2i^2+ x2v = 2x2i^  
  • For example with x = 3
    • (3^ +3i^) = 9^ + 9i^ + 9i^ + 9v = 18i^ = 2*32i^

  • (xv  + xiv)2  = x2^ + 2x2i^+ x2v = 2x2i^
  • For example with x = 3
    • (3v +3iv) = 9^ + 9i^ + 9i^ + 9v = 18i^ = 2*32i^

As a result √(2x2i^ ) = (x^  + xi^)2 or (xv  + xiv)2

Calculate the full roots as the square root of the Counter times the square root of the Unitary.

  • √(2x2i^ ) = √2*√x2*√i^ = √2x(√2/2^ + √2/2i^)
    • = 2x/2^  + 2x/2i^
    • = x^ + xi^
    • For example for x = 3
      • √(2*32i^ ) = √2*√32*√i^ = √2*3(√2/2^ + √2/2i^) = 3^ + 3i^ = x^ + xi^ 

or

  • √(2x2i^ ) = √2*√x2*√i^ = √2x(√2/2v + √2/2iv)
    • = 2x/2v  + 2x/2iv
    • = xv + xiv 
    • For example for x = 3
      • √(2*32i^ ) = √2*√32*√i^ = √2*3 (√2/2v + √2/2iv) = 3v  + 3iv = xv + xiv

Conclusion

Finally, try these examples of the R2 roots of expressions with the help of our online calculator.

 

Next: Advanced Roots

Previous: Simple Roots

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