Question

What is \[{{i}^{3}}\]?

Answer 1

Hint: In the question that has been stated above where ever you see \[i\] in mathematics always remember that it is none other than complex number \[i\] which indicates square root of -1 i.e. \[\sqrt{-1}\], now that we have known this we need to find the value of \[{{\left( \sqrt{-1} \right)}^{3}}\].

Complete step by step answer:
In the above stated question \[i\] signifies the complex number \[i\] which is none other than\[\sqrt{-1}\], now when we say that we need cubic value of a square we multiply it individually and when we does this, we know that when we multiply two square roots at of same base value the result is out as base value, so when we do the same with complex number \[i\] i.e. multiply \[i\times i\] which is none other than \[{{i}^{2}}\] we will get the resultant as -1 which is the base value of \[i\] and as described above that the multiplication of two same values under square root give the value as output. So now when we multiply the third \[i\] to \[{{i}^{2}}\] which is \[-1\times i\] we will get the resultant\[-i\]. so when we multiply \[i\]three times to itself we will get a resultant of \[-i\] which we can also write as –square root -1 i.e. \[-\sqrt{-1}\].

The value that we get when \[i\] is multiplied to itself thrice i.e.\[{{i}^{3}}\] , we get the final product as \[-i\] or \[-\sqrt{-1}\].

Note: Now in the above stated question it becomes sometimes confusing what the value is going to come out, so try to multiply the first few i terms so that we can get a series trend and then we can use the same trend series to get further values.