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

Question
Answer 1

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

#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#

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Answer 2

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

#= 512/19683#

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#

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