How do you evaluate the expression #(2/3)^4/((2/3)^-5(2/3)^0)# using the properties of indices?.

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Answer 1 · 2017-05-23T14:32:23

#(2/3)^4/((2/3)^-5(2/3)^0)=(2/3)^9=512/19683#

We can use here the identities

#a^mxxa^n=a^((m+n))# and #a^m/a^n=a^((m-n))#

As such #a^m/(a^na^p)=a^((m-n-p))#

Hence #(2/3)^4/((2/3)^-5(2/3)^0)#

#=(2/3)^((4-(-5)-0))#

#=(2/3)^((4+5))#

#=(2/3)^9#

or #2^9/3^9=512/19683#

Answer 2 · 2017-05-23T14:38:44

#2^9/3^9 = (2/3)^9#

#= 512/19683#

There are four properties of indices (exponents) to consider here:

Raising factors to a power: #color(blue)((xy)^m = x^m xx y^m)#

A negative index: #color(magenta)(1/x^-m = x^m)#

Index of #0#. #color(lime)("Anything to power of 0 is equal to"1)" "#(except #0^0#)

Multiply law: same bases, add the indices: #x^m xx x^n = x^(m+n)#

#color(blue)((2/3)^4)/(color(magenta)((2/3)^-5)color(lime)((2/3)^0)) = (color(blue)(2^4/3^4)xxcolor(magenta)((2/3)^5))/(color(lime)((1))#

#=2^4/3^4xx2^5/3^5#

#= 2^9/3^9#

This can also be written as #(2/3)^9#

#= 512/19683#