How do you simplify #(a^4)^-5 * a^13#?

1 Answer
Zor Shekhtman
May 17, 2015

By definition, #x^(-N)=1/(x^N)#.
Therefore,
#(a^4)^(-5)=1/((a^4)^5)#

By definition. #(x^M)^N = (x^M) * (x^M) * (x^M)...#
(where multiplication is performed #N# times)
But each #x^M = x*x*x...#
(where multiplication is performed #M# times)
Therefore,
#(x^M)^N = x*x*x*...#
(where multiplication is performed #M*N# times).
Hence, #(x^M)^N=x^(M*N)#

Using the above, we can write:
#(a^4)^(-5)=1/((a^4)^5)=1/a^(4*5)=1/a^20#

The original expression, therefore, equals to
#(a^4)^(-5)*a^13=a^13/a^20=1/a^7=a^(-7)#

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