Is $5 = -5$ according to this formula?

Question

Today in maths class something occurred to me. How can this be:

$\sqrt{(-5)^2} = ((-5)^2)^{1/2} = (-5)^{2(1/2)} = (-5)^1 = -5$
$\sqrt{(-5)^2} = ((-5)^2)^{1/2} = 25^{1/2} = 5$

This could theoretically prove that $5$ equals $-5$.

Is there any logical reason to explain this?

This kind of exponentiation only works for real numbers greater than or equal to zero. There is some funny business when you try to exponentiate negative numbers. — Em., Feb 09 '16 at 22:48
Related: http://matheducators.stackexchange.com/questions/10488/student-converted-sqrtx2-and-ended-up-with-just-x-instead-of-x — mweiss, Feb 09 '16 at 22:54

score 1 · Accepted Answer · answered Feb 09 '16 at 23:02

1

Technically, the $\sqrt{x^2}$ could be $±x$ because $x^2 = (-x)^2,$ but the convention is to only use the positive root for reasons elaborated in the links Jack's wasted life and mweiss provided.

answered Feb 09 '16 at 23:02

Simpson17866

522

Is $5 = -5$ according to this formula?

1 Answers1