How do you simplify #(8/27)^(-2/3)#?

2 Answers
Mia
Feb 14, 2017

#9/4#

Explanation:

#(8/27)^(-2/3)#
#" "#
#=(27/8)^(2/3)#
#" "#
The prime factorization of 27 and 8 is:
#" "#
#27=9xx3=3xx3xx3=3^3#
#" "#
#8=4xx2=2xx2xx2=2^3#

Substituting the factorization on the above fraction we have:
#" "#
#(27/8)^(2/3)#
#" "#
#=(3^3/2^3)^(2/3)#
#" "#
#=((3/2)^3)^(2/3)#
#" "#
Applying the property of power of a power that says:
#" "#
#color(red)((a^n)^m=a^(mxxn))#
#" "#
#((3/2)^3)^(2/3)#
#" "#
#=(3/2)^(3xx(2/3))#
#" "#
#=(3/2)^2#
#" "#
#=9/4#

EZ as pi
Feb 14, 2017

#=9/4#

Explanation:

There are 3 different processes indicated in this expression.

Laws of indices:

#x^-m = 1/x^m" "and" "(a/b)^-m = (b/a)^m#

The second law is the one we will apply.

Also #x^(p/q) = rootq(x)^p#

#(8/27)^(-2/3) = (27/8)^(+2/3)#

#= root3(27/8)^2" "larr# find the cube roots first

#=(3/2)^2#

#=9/4#

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