Why is it when you square root a number you get a +/-? Why is it when you solve $\displaystyle -4^2$ it is -16 and not 16? How can you tell it is +/- for the second question?
Hi there
$\displaystyle -4^2=-(4)(4)=-16$
but
$\displaystyle (-4)^2=(-4)(-4)=16$
You get a $\displaystyle \pm$ when you take the square root of a number because both the positive and negative square roots could be the factors: e.g. $\displaystyle 4\times4=16$ and $\displaystyle (-4)\times(-4)=16$
$\displaystyle -(4)^2=-(-4)^2=-16$
$\displaystyle (4)^2=16,\ -(4^2)=-16$
$\displaystyle (-4)^2=16,\ -(-4)^2=-16$
$\displaystyle 4(4)=4+4+4+4=16$
$\displaystyle -4(4)=-4-4-4-4=-16$
$\displaystyle Hence,\ negative\ by\ positive\ means\ successive\ subtraction.$
$\displaystyle (-4)(-4)=-[4(-4)]=\ opposite\ of\ subtract\ 4\ four\ times,$
$\displaystyle which\ is\ add\ 4\ four\ times.\ Add\ and\ subtract\ are\ opposites.$
$\displaystyle This\ is\ why\ negative\ by\ negative\ is\ really\ positive\ by\ positive.$
$\displaystyle To\ write\ -4^2\ is\ ambiguous.$
$\displaystyle We\ must\ distinguish\ between\ -[(4)^2]\ and\ [(-4)^2]$