Multiply the given below and write the answer in standard form.

(2 - √-100 )(1 + √-36 )

Solution

If we have to multiply this out in its present form we would get,

 (2 -     √-100 )(1 +     √-36 ) = 2 + 2√-36 - √-100 -√-36 √-100

Now, if we were not being careful we would possibly combine the two roots into the final term into one that can't be done!

Thus, there is general rule of thumb in dealing along with square roots of negative numbers. While faced with them the first thing which you have to always do is convert them to complex number.  If we follow this rule we will always acquire the correct answer.

So, let's work on this problem the way it have to be worked.

(2 -√-100 )(1+√-36 ) = ( 2 -10i ) (1 + 6i ) = 2 + 2i - 60i²  = 62 + 2i

The rule of thumb given in the earlier example is important adequate to make again.  While faced with square roots of negative numbers the first thing which you have to do is convert them to complex numbers.

There is one final topic that we have to touch on before leaving this section. Since we noted on radicals even though √9 = 3 there are in fact two numbers that we can square to obtain 9.  We can square 3 and -3 both.

The similar will hold for square roots of -ve numbers. As we saw earlier √-9 = 3i .  As with

Square roots of positive numbers in this case actually we are asking what did we square to acquire -9? Well it's simple enough to check that 3i is correct.

                                              (3i )²  = 9i²  = -9

Though, i.e. not the only possibility. 

Consider the following,

                                      ( -3i )²  = ( -3)² i²= 9i²  = -9

and thus if we square -3i we will also acquire -9.  Thus, when taking the square root of a negative number there are actually two numbers which we can square to get the number under the radical.  Though, we will always take the positive number for the value of the square root just as we do along with the square root of positive numbers.