How do you evaluate #(\frac { p ^ { - 1/ 2} q ^ { - 5/ 3} } { 2^ { - 1} p ^ { - 3} q ^ { - 1/ 5} } ) ^ { - 3}#?

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Answer 1 · 2018-04-29T17:18:19

See below

Apply power rules #(a^n)^=a^(mn)# and #(a/b)^n=a^n/b^n#

#p^((-1/2)(-3))q^((-5/3)(-3))=p^(3/2)q^5# in numerator

#2^((-1)(-3))p^((-3)(-3))q^((-1/5)(-3))=2^3p^9q^(3/5)# denominator

Now the quotient applying #a^m/a^n=a^(m-n)#

#(p^(3/2)q^5)/(2^3p^9q^(3/5))=2^(-3)p^(3/2-9)q^(5-3/5)^=2^(-3)p^(-15/2)q^(22/5)#

We can stop here or develope prior expresion applying #a^(m/n)=root(n)(a^m)#

#root(5)(q^22)/(2^3sqrt(p^15))=(q^4root(5)(q^2))/(2^3p^7sqrtp)#