Question

What types of numbers are undefined when they are under a radical sign? If you were dealing with the number √-1, would it be defined if you multiplied it by 2? Would it be defined if you subtracted some real number from it? Would it be defined if you squared it? Would it be defined if you cubed it? Think about raising this number to any positive integer. When would the result be defined? When would the result not be defined? What method would you use to square the value 3 + √-1 ? In what other cases would you use this method? Explain.

Answer 1

1) the types of number are the negative integers (e.g √-1 √-3 √-5 are not defined)
2) the answer is No, proof:  2x√-1 is not defined because √-1 doesn't exist
3) the answer is No, proof:  √-1 - 3 is not defined because √-1 doesn't exist
4) the answer is Yes, proof: (√-1 )²=  -1 this is a real number
5) the answer is No, proof:  (√-1 )^3=  (√-1 )²(√-1 )= - 1(√-1 ), and - 1(√-1 ) is not defined because √-1 doesn't exist
6) the result would be defined with the following cases:
    √-1+n,  n>1
     √-1xn,  n<0
     √-1/n,   n<0
7) the result would not be defined with the following cases:
   √-1+n,  n<0
     √-1xn,  n>0
     √-1/n,   n>0
 8) to square 3 + √-1, I use the method of complex number
  i²= -1, it implies i= √-1
  so 3+√-1=3+i,  and then (3+√-1)²=(3+i)²= 9 -1+6i= 8-i= 8-√-1
 9) it is used for finding complex roots of a number