sqrt(-4) = (-4)^(1/2) = ((-4)^(2))^(1/4) = sqrt((-4)^(2),4) = sqrt(16,4) = 2. Where is mistake?

vasiakriuk Apr 30, 2014

#2**+10 **

You have made use of the relation x^{a*b} = (x^{a})^{b}. But this has a limited range of applicability. In particular it doesn't apply when x is negative and b is a fraction. If you insist on applying it outside of its range of applicability you will get inconsistent mathematics (as your example illustrates!).

Alan Apr 30, 2014

#1**0 **

sqrt(-4) = (-4)^(1/2) = ((-4)^(2))^(1/4) = sqrt((-4)^(2),4) = sqrt(16,4) = 2. Where is mistake?

-----------------------------------------------------------------------------------------------------

Note that √(-4) = √(4)*√(-1) =2√(-1)

And by definition, √(-1) = √(i^{2}) = i

So, √(-4) = 2i ≠ 2

CPhill Apr 30, 2014

#2**+10 **

Best Answer

You have made use of the relation x^{a*b} = (x^{a})^{b}. But this has a limited range of applicability. In particular it doesn't apply when x is negative and b is a fraction. If you insist on applying it outside of its range of applicability you will get inconsistent mathematics (as your example illustrates!).

Alan Apr 30, 2014

#3**+5 **

The mistake lies partly in writing 1/2 as 2/4. Squaring an expression often results in the creation of spurious answers, as for example, x=2, x^2=4, x=sqrt(4), x=2 or x=-2.

As to the actual question, to discuss it you have to involve complex numbers.

$$(-4)^{1/2}=(4\angle(180+360k))^{1/2}=2\angle(90+180k),\quad k=0,1.$$

That gets you 2 values for the square root of -4,

$$2\angle 90=2i \text{ or } 2\angle 270=-2i.$$

Now look at the final result as given, sqrt(16,4), which I take to mean the fourth root of 16.

$$16^{1/4}=(16\angle(0+360k))^{1/4}=2\angle 90k, \quad k=0,1,2,3.$$

That produces 4 'answers',

$$2\angle 0=2,\quad 2\angle90=2i,\quad 2\angle 180=-2 \quad \text{ and }\quad 2\angle 270=-2i,$$

that is, the two correct values plus two spurious ones.

The second mistake is that of choosing one of the spurious answers as the answer !

Bertie Apr 30, 2014