First, we can use this rule for exponents to eliminate the negative exponent:
#x^color(red)(a) = 1/x^color(red)(-a)#
#1000^color(red)(-2/3) = 1/1000^color(red)(- -2/3) = 1/1000^color(red)(2/3)#
We can next rewrite the exponent as:
#1/1000^color(red)(2/3) = 1/1000^(1/3 xx 2)#
And then use this rule to rewrite the expression again:
#x^(color(red)(a) xx color(blue)(b)) = (x^color(red)(a))^color(blue)(b)#
#1/1000^(color(red)(1/3) xx color(blue)(2)) = 1/(1000^(color(red)(1/3)))^color(blue)(2)#
We can convert the term within the parenthesis to radical form using this rule:
#x^(1/color(red)(n)) = root(color(red)(n))(x)#
#1/(1000^(color(red)(1/3)))^color(blue)(2) = 1/(root(color(red)(3))(1000))^color(blue)(2) = 1/10^color(blue)(2) = 1/100#
