R3 – Square Roots

Definition

The square root of an expression in R3 is made of the roots of the individual Opposite Values with the signs remaining constant.

Square of an Expression

Given the generic expression (a^? + bi^? + cj^?):

• (a^? + bi^? + cj^?)² = a^2? +  a^?*bi^? + a^?*cj^? + bi^?*cj^? + bi^?*a^? + b²i^? + cj^?*bi^? + c²j^? + cj^?*a^?   

Note that as a result of the Zero Rules of Rotation a?^?b?^? + b?^?a?^? = 0, so

• (a^? + bi^? + cj^?)² = a^?2 +  bi^?2 + cj^?2   

This shows that the square of an expression is the square of the individual Opposite Values, therefore the square root of an expression is the square root of the individual Opposite Values.

R3 Square Root Example

Firstly, square (3^v + 4i^{^} + 5j^v) using long multiplication. 

3^v + 4i^{^} + 5j^v    

3^v + 4i^{^} + 5j^v    

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  9^v + 12j^v  + 15i^v    

20^v + 12j^{^}  +  16i^{^}   

20^{^} + 25j^v   + 15i^{^}     

==============   

9^v   + 16i^{^} + 25j^v     

The long multiplication shows that:

• (3^v + 4i^{^} + 5j^v )*(3^v + 4i^{^} + 5j^v ) = (9^v + 16i^{^} + 25j^v ) = (3^v + 4i^{^} + 5j^v )² = (3^v2 + 4i^{^2} + 5j^v2 )
• Therefore,  √(9^v + 16i^{^} + 25j^v ) = (√9^v + √16i^{^} + √25j^v ) = (3^v + 4i^{^} + 5j^v )

Finding the R3 Square Root using Formula  

The square root of any equation is easily solved using the roots of the individual Opposite Values:

• (a^? + bi^?+ cj^?)*(a^? + bi^?+ cj^?) = (9^v + 16i^{^} + 25j^v)
• (a^? + bi^?+ cj^?) = (√9^v + √16i^{^} + √25j^v)
• (a^? + bi^?+ cj^?) = (3^v + 4i^{^} + 5j^v)

Conclusion

Try R3 square roots with our online calculator.

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