How do you simplify #16^(-2/3)#?

1 Answer
Apr 5, 2018

#color(blue)(16^(-2/3) = 1/(4*2^(2/3#

Explanation:

Given:
#" "#
#color(blue)(16^(-2/3)#

Identities used:

#color(red)(a^(-b)=1/(a^b)#

#color(red)(a^(m/n) = rootn(a^m)#

Consider the expression given:

#color(blue)(16^(-2/3)#

#rArr 1/(16^(2/3))#

#rArr 1/root3(16^2)#

#rArr 1/root3(256#

#rArr 1/root3(64*4)#

Note that #color(blue)(64 = 4^3#

#rArr 1/root3(4^3*4)#

#rArr 1/root3(4^3)*1/root3(4)#

Note that #color(red)(rootp(m^n) = (m^n)^(1/p)#

#rArr 1/(4^3)^(1/3)*1/root3(4)#

#rArr (1/4)*1/root3(4)#

#rArr (1/4)*1/root3(2^2)#

Note that #color(red)(sqrt(m^n) = (m^n)^(1/2)=m^(n/2)#

#rArr 1/4*[1/[(2^2)^(1/3))]#

#rArr 1/4*1/(2^(2/3))#

#rArr 1/(4*2^(2/3)#

Hence,

#color(blue)(16^(-2/3) = 1/(4*2^(2/3)#

If you wish, you can continue to simplify further:

#1/(4*2^(2/3)#

#rArr 1/(2^2*2^(2/3)#

Note that #color(red)(a^m*a^n=a^(m+n)#

#rArr 1/2^(2+(2/3)#

#rArr 1/(2^(8/3)#

#rArr 2^(-8/3#

Hope you find this solution useful.