Since √a√b=√ab, consider the following string of equalities:

−1=(i)(i)=√−1√−1=√(−1)(−1)=√1=1

There seems to be something wrong here. What is it?

Question

Since √a√b=√ab, consider the following string of equalities:

−1=(i)(i)=√−1√−1=√(−1)(−1)=√1=1

There seems to be something wrong here. What is it?

anonymous · Answer

@calculusfunctions

aravindg · Answer

Well √a√b=√ab is an incomplete sentence .

It is ONLY valid when a>o and b >o

aravindg · Answer

For cases where a<o and b<0 the rule is 

√a√b=-√ab

aravindg · Answer

Hope you understand

anonymous · Answer

Okay. But I think with that information aside, we're just trying to figure out whats wrong with the string of equalities @AravindG

anonymous · Answer

@Hunus Can you help me? I've seen you typing for a while.

aravindg · Answer

Nope you see no one said that you could apply   √a√b=√ab for ALL cases .It is ONLY true when a>0 and b>0 .Infact √a√b=√ab is a false statement unless its followed by the condition a>0 and b>0

hunus · Answer

Yes, the square root of a function geometrically says if you have two squares one with area a and the other with area b. 

What are the side lengths of those areas?

|dw:1367897998851:dw|

$$\sqrt{a}$$
and

$$\sqrt{b}$$
So the square root of -1 is asking, given a square with area -1, what is the side length?

Which in and of itself makes no sense geometrically.

Once a and b are outside the realm of things we can physically perceive and understand (i.e. Shapes with negative side length or negative area) We can't use the same methods we've been using.

It happens to turn out that 'if' a and b are less than zero, then the square root of a times the square root of b is equal to the negative of the square root of a times b

$$\sqrt{a}\sqrt{b} = -\sqrt{ab}$$ for negative values of a and b

Geometrically it turns out that the side length of a square times the side length of another square is the same as the side length of a square with an area that is given by the multiplication of the areas of the original two squares.

$$\sqrt{a}\sqrt{b} = \sqrt{ab}$$ for values of a and b that are greater than or equal to zero

This is a geometrical happenstance. 

Now, a shape with negative area doesn't have any physical significance and the geometrical happenstance doesn't work the same way.

hunus · Answer

|dw:1367900249964:dw|

This should be square root a and square root b