R2 – Exponentiation – Fractions, Flip Sign and Rules

This post covers the rules of R2 exponentiation. It also describes the use of fractions and the flip sign with exponentiation along with some examples.

Fractions

Exponents in fractions represent multiplication roots. As described in an earlier post, the root of an expression can be expressed as the root of the counter multiplied by the root of the unitary. For example:

• 10^{^1/2} = 10^1/2*1^{^1/2}  =  ²√10*²√(1^{^}) = 3.16*1^{^} = 3.16^{^}  
  • or 3.16*1^v = 3.16^v

• 10^v1/2 = 10^1/2*1^v1/2  =  ²√10*²√(1^v) = 3.16*i^{^} = 3.16i^{^}  
  • or 3.16*i^v = 3.16i^v

• 10^{^2/3} = 10^2/3*1^{^2/3}  =  (³√10)²*³√(1^{^})² = (2.154)²*(1^{^})² = 4.64^{^}
  • or (2.154)²*(0.5^v + 0.806i^{^})² = 4.64*(0.4^v + 0.806i^v) = 1.855^v + 3.74i^v
  • or (2.154)²*(0.5^v + 0.806i^v)² = 4.64*(0.4^v + 0.806i^{^}) = 1.855^v + 3.74i^{^}

• 10^v2/3 = 10^2/3*1^v2/3  =  (³√10)²*³√(1^v)² = (2.154)²*(1^v)² = 4.64^{^}
  •  or (2.154)²*(0.5^{^} + 0.806i^{^})² = 4.64*(0.4^v + 0.806i^{^}) = 1.855^v + 3.74i^{^}
  • or (2.154)²*(0.5^{^} + 0.806i^v)² = 4.64*(0.4^v + 0.806i^v) = 1.855^v + 3.74i^v

• 10i^{^1/2} = 10^1/2*i^{^1/2}  =  ²√10*²√(i^{^}) = 3.16*(√2/2^{^} + √2/2i^{^}) =2.234^{^} + 2.234i^{^}
  • or 3.16*(√2/2^v + √2/2i^v)  =1.58√2^v + 1.58√2i^v

• 10i^v1/2 = 10^1/2*i^v1/2  =  ²√10*²√(i^v) = 3.16*(√2/2^{^} + √2/2i^v) =2.234^{^} + 2.234i^v
  • or 3.16*(√2/2^v + √2/2i^{^})  =2.234^v + 2.234i^{^}

*Notes:

• Some basic square roots, such as ²√i^v are in the post on simple multiplication roots.

The exponent can be applied to the counter only as follows:

• 10^{1/2^} = ²√10^{1^} = 3.16^{1^}  = 3.16^{^} 
• 10^1/2v = ²√10^1v = 3.16¹i^v = 3.16i^v
• 10^{2/3^} = ³√10^{2^} = 2.154^{2^} = 4.64^{^}   
• 10^2/3v = ³√10^2v = 2.154^2v = 4.64^v                            
• 10^{3/4^} = ⁴√10^{3^} = 1.778^{3^} = 5.62^{^}

Exponent with Flip Sign and Fractions as Reciprocals of Roots

Exponents with a flip sign before a fraction represents the reciprocals of roots. For example:

• 10^{^-2/3} = 1^{^}/10^{^2/3} = 1^{^}/(10^2/3*1^{^2/3}) = 1^{^}/(³√10²*³√(1^{^})²) = 1^{^}/(2.154²*1^{^}) =1^{^}/4.64^{^} = 0.216^{^}
  • or 1^{^}/(2.154²*((0.5^v + 0.866i^{^})²) =1^{^}/(4.64*(0.5^v + 0.866i^v)) = 1^{^}/(2.32^v + 4.02i^v)= 0.108^v +0.187i^{^}
  • or 1^{^}/(2.154²*((0.5^v + 0.866i^v)²) =1^{^}/(4.64*(0.5^v + 0.866i^{^})) = 1^{^}/(2.32^v + 4.02i^{^})= 0.108^v +0.187i^v

• 10^v-2/3 = 1^{^}/10^v2/3 = 1^{^}/(10^2/3*1^v2/3) = 1^{^}/(³√10²*³√(1^v)²) = 1^{^}/(2.154²*1^v) =1^{^}/4.64^v = 0.216^v
  • or 1^{^}/(2.154²*((0.5^{^} + 0.866i^{^})²) =1^{^}/(4.64^{^}*(0.5^v + 0.866i^{^})) = 1^{^}/(2.32^v + 4.02i^{^})= 0.108^v +0.187i^v
  • or 1^{^}/(2.154²*((0.5^{^} + 0.866i^v)²) =1^{^}/(4.64^{^}*(0.5^v + 0.866i^v)) = 1^{^}/(2.32^v + 4.02i^v)= 0.108^v +0.187i^{^}

• 10i^{^-1/2} = 1^{^}/10i^{^1/2} = 1^{^}/(10^1/2*i^{^1/2}) = 1^{^}/(√10*√(i^{^})) = 1^{^}/(3.16*(0.707^{^} + 0.707i^{^})) =1^{^}/(2.234^{^} + 2.234i^{^}) = 0.224^{^} + 0/224i^v
  •  or 1^{^}/(3.16*(0.707^v + 0.707i^v)) =1^{^}/(2.234^v + 2.234i^v) = 0.224^v + 0.224i^{^}

• 10i^v-1/2 = 1^{^}/10i^v1/2 = 1^{^}/(10^1/2*i^v1/2) = 1^{^}/(√10*√(i^v)) = 1^{^}/(3.16*(√2/2^{^} + √2/2i^v)) =1^{^}/(2.234^{^} + 2.234i^v) = 0.224^{^} + 0.224i^{^}
  •  or 1^{^}/(3.16*(√2/2^v + √2/2i^{^})) =1^{^}/(2.234^v + 2.234i^{^}) = 0.224^v + 0.224i^v

The exponent can be applied to the counter only as follows:

• 10^{-1/2^} = 1/(²√10)^{1^} = 1/(3.16)^{^} = 0.316^{^}
• 10^-1/2v = 1/(²√10)^1v = 1/(3.16)^v = 0.316^v
• 10^{-2/3^} = 1/(³√10)^{2^} = 1/(2.154)^{2^} = 1/4.64^{^} = 0.216^{^}
• 10^-2/3v = 1(³√10)^2v = 1/(2.154)^2v = 1/4.64^v = 0.216^v
• 10^{-3/4^} = 1/(⁴√10)^{3^} = 1/(1.778)^{3^} = 1/5.62^{^} = 0.178^{^}

Flipping

The flip sign can also be used in exponentiation as a flipped number can be raised exponentially. For example:

• (^–2^{^})² = (2^v)² =  2^v*2^v = 4^{^}
• (^–2^v)² = (2^{^})² =  2^{^}*2^{^} = 4^{^}
• (^–2i^{^})³ = 2i^v*2i^v*2i^v = 8i^{^}                                          
• (^–2i^v)³ = 2i^{^}*2i^{^}*2i^{^} = 8i^v
• (^–2i^v)⁴ = 2i^{^}*2i^{^}*2i^{^}*2i^{^} = 16^{^}

It can make a difference which of the flip and exponential operations takes place first, so the flip sign can produce a different result if outside the brackets. For three of the examples above:

• ^–(2^{^2})  = ^–(2^{^}*2^{^}) = ^–(4^{^}) = 4^v. Note that this is different to (^–2^{^})² above.
• ^–(2i^v3)  = ^–(2i^v*2i^v*2i^v) = ^–(8i^{^}) = 8i^v. Note that this is the same as (^–2i^v)³ above.
• ^–(2i^v)⁴ = ^–(2i^v*2i^v*2i^v*2i^v)= ^–(16^{^}) = 16^v. Note that this is different to (^–2^v)⁴ above.

Finally for flipping:

• ^–2^v2 is interpreted as ^–(2^v2) = ^–(4^{^}) = 4^v and
• ^–2i^v2 is interpreted as ^–(2i^v2) = ^–(4^v) = 4^{^}

R2 Rules of Exponentiation

There are 14 R2 rules of exponentiation as follows:

1. xⁿ*x^m      = x^n+m
2. xⁿ/x^m     = x^n+–m, when x ≠ 0                       
3. (xⁿ)^m       = x^nm
4. (xy)ⁿ       = xⁿyⁿ
5. (x/y)ⁿ     = xⁿ/yⁿ , when y ≠ 0
6. x^–n           = 1^{^}/xⁿ, when x ≠ 0                                
7. √x           = x^1/2
8. x⁰            = 1^{^}, when x ≠ 0   
9. 0⁰           = 0         
   • Note that it has not been agreed in classical maths whether 0⁰ = 1 or is undefined.
10. x^2/3         = (³√x)²
11. x^m/n = x^m/n*1‡x^‡xm/n
    • For example 10^v1/2 = 10^1/2*1^v1/2 
12. (xyz)³     = x³y³z³
13. ³√⁴√a     = ¹²√a
14.  (x + y)²  ≠ x² + y²

Conclusion

Finally, an exponent is not an Opposite Value. It is a Counter. The online calculator does not support exponentiation yet.

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