R1 – Exponentiation – Fractions, Flip Sign and Rules

This post covers the use of exponents with fractions and flip signs as well as the rules of exponentiation in R1.                                         

Exponent Fractions as Roots

Exponents in fractions represent 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} = ²√10*²√(1^{^})¹ =3.16*1^{^} = 3.16^{^}
  • or 3.16*1^v = 3.16^v

• 10^v1/2 is N/A as square roots of ^v Opposite Values are not allowed in R1

• 10^{^2/3} = ³√10²*³√(1^{^})² = 2.154²*1^{^2} = 4.64^{^}

• 10^v2/3 = ³√10²*³√(1^v)² = 2.154²*1^v2 = 4.64^{^}
  • Note that this is the same value as for 10^{^2/3}. This reflects that the square root of a ^{^} Opposite Value can be either ^{^} or ^v.

• 10^{^3/4} = ⁴√10³*⁴√(1^{^})³ = 1.778³*1^{^3} = 5.62^{^} or
  • 1.778³*1^v3 = 5.62^v

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

• 10^{1/2^} = ²√10^{1^} = 3.16^{^} = 3.16^{^}
• 10^1/2v = ²√10^1v = 3.16^v = 3.16^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^{^-1/2} = 1^{^}/10^{^1/2} = 1^{^}/(²√(10^{^})¹) = 1^{^}/(3.16^{^})¹ = 0.316^{^}
  • or 1^{^}/(3.16^v)¹ = 0.316^v

• 10^v-1/2 is N/A as square roots of ^v Opposite Values are not allowed in R1

• 10^{^-2/3} = 1^{^}/10^{^2/3} = 1^{^}/(³√(10^{^})²) = 1^{^}/(2.154^{^})² = 1^{^}/4.64^{^}  = 0.216^{^}

• 10^v-2/3 = 1^{^}/10^v2/3 = 1^{^}/(³√10^v)² = 1^{^}/(2.154^v)² = 1^{^}/4.64^{^} = 0.216^{^}
  • Note that this is the same value as for 10^{-^2/3}. This reflects that the square root of a ^{^} Opposite Value can be either ^{^} or ^v  

• 10^{^-3/4} = 1^{^}/10^{^3/4} = 1^{^}/(⁴√10^{^})³ = 1^{^}/(1.778^{^})³ = 1^{^}/5.62^{^} = 0.178^{^}
  • or 1^{^}/(1.778^v)³ = 1^{^}/ 5.62^v = 0.178^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^{^}

Exponents with Flipped Opposite Values

Exponents can be used with flipped Opposite Values in R1. For example:

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

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

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

^–2^{^2} is interpreted as ^–(2^{^2}) and equals 4^v.

^–2^v2 is interpreted as ^–(2^v2) and equals 4^v.

Finally, an exponent is not an Opposite Value. It is a Counter.

Rules of R1 Exponentiation

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. (xyz)³     = x³y³z³
12. ³√⁴√a     = ¹²√a
13.  (x + y)²  ≠ x² + y²

Next: Logs Definition

Previous: Exponentiation Definition