R3 – Exponentiation – Fractions, Flip Sign and Rules

This post covers the rules of R3 exponentiation. It also describes the use of fractions and flip sign with exponentiation along with some examples.

Fractions

Exponents in fractions represent multiplication roots. The post on R3 roots outlined how the Counter and Full roots of Opposite Values are always equal and how there is only 1 root in R3 for an expression.

1. 10^1/2j^{^} = (²√10j^{^})¹ = (3.16j^{^})¹  = 3.16j^{^} 
2. 10^1/2j^v = (²√10j^v)¹ = (3.16j^v)¹ = 3.16j^v 
3. 10^2/3i^{^} = (³√10i^{^})² = (2.154i^{^})² = 4.64i^{^}   
   • Note: 4.64i^{^3} = 100i^{^} = 10i^{^2}
4. 10^2/3i^v = (³√10i^v)² = (2.154i^v)² = 4.64i^v                            
5. 10^{3/4^} = (⁴√10^{^})³ = (1.778^{^})³ = 5.62^{^} 
6. 10^3/4v = (⁴√10^v)³ = (1.778^v)³ = 5.62^v
7. 10^{2^}/10²i^v = 100^{^}/100i^v = j^v = 10^{(2+–2)(^/iv)} = 10⁰j^v = j^v
8. 10^1/2v = (√10^v)¹ = (3.16^v)¹ = 3.16^v 
9. 10^1/2i^{^} = (√10i^{^})¹ = (3.16i^{^})¹ = 3.16i^{^}
10. 10^1/2i^v = (√10i^v)¹ = (3.16i^v)¹ = 3.16i^v
11. 10^3/4i^v = (⁴√10i^v)³ = (1.778i^v)³ 5.62i^v

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:

• 10j^{^-2/3} = j^{^}/10^2/3 = j^{^}/(³√10²) = j^{^}/(2.154²) =j^{^}/4.64^{^} = 0.216j^{^}

• 10j^v-2/3 = j^v/10^2/3 = j^v/(³√10²) = j^v/(2.154²) = j^v/4.64 = 0.216j^v  

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

• 10i^v-1/2 = i^v/10^1/2 = i^v/(√10) = i^v/3.16 =0.316i^v

Flipping

The flip sign can also be used in exponentiation:

A flipped number can be raised exponentially.

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

It makes no difference which of the flip and exponential operations takes place first, so the flip sign produces the same result if outside the brackets. For the last two examples above:

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

An exponent is not an Opposite Value. It is a Counter.

R3 Rules of 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^‡x/|xⁿ|, when x ≠ 0                              
7. √x           = x^1/2
8. x⁰            =  1^‡x, 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 + z)²  = (x² + y² + z²)

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