Any (non-zero) number to the power of zero equals #1#

So #1/color(red)(2^0) = 1/color(red)(1) = 1#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**What is the logic behind "any number to the power of zero equals #1#"?**

#a^n=a^(n-1)xxa#

#color(white)("XXX")#that is #a# multiplied by itself #n# times

#color(white)("XXX")#is the same as

#color(white)("XXXXXX")a# multiplied together #(n-1)# times

#color(white)("XXXXXX")#and then multiplied by #a# one more time.

This could be re-written as

#a^(n-1)=a^n div a# (provided #a!=0#)

So

#color(white)("XXX")a^2=a^3div a#

#color(white)("XXX")a^1=a^2diva#

#color(white)("XXX")a^0=a^1diva#

but

#color(white)("XXX")a^1=a#

and therefore

#color(white)("XXX")a^0=adiv a=1# (again, provided #a!=0#)