Question 759572
-8^(-5/3) = -1/(8^(5/3)) = -1/(8^(1/3))^5 = -1/32
{{{-8^(-5/3) = -1/((8^(5/3))) = -1/(8^(1/3))^5 = -1/32}}}
 
The first two equal signs are just the definition of negative exponent and the definition of rational exponent (with a minus sign in front of each expression).
I cannot imagine any intermediate steps there.
Definition of rational exponent (for positive {{{b}}} and positive integers {{{m}}} and {{{n}}}):
{{{b^(m/n)=(b^(1/n))^m}}} or {{{(root(n,b))^m}}} or {{{(b^m)^(1/n)}}} or {{{root(n,b^m)}}}
I cannot imagine any intermediate steps there.
Definition of negative exponent (for any {{{x}}} if {{{b}}} is positive):
{{{b^(-x)=1/b^x}}}
 
The last equal sign could be explained by
{{{8=2^3}}}
{{{root(3,8)=8^(1/3)=(2^3)^(1/3)=2^((3*(1/3)))=2^1=2}}}
and {{{2^5=32}}}
You could write out more in-between steps there, but I would think that anything more than
... = -1/(8^1/3)^5 = 1/2^5 = -1/32
is overkill.