How to find the value for the expression ?

Question

Given #2^n = 9# , find the value for the expression :

#(8^(1/3n)*2^(5/2n))/4^n#

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Answer 1 · 2017-09-01T17:20:10

# 27.#

The Expression #={(8^(1/3*n))(2^(5/2*n))}/(4^n)......(1).#

We know that, #(a^p)^q=a^(pq).#

#:. 8^(1/3*n)=(2^3)^(1/3*n)=2^(3*1/3*n)=2^n.................(2), and, #

#4^n=(2^2)^n=2^(2n)................(3).#

Using #(2), and, (3)" in "(1),# we have,

#"The Exp.="(2^n*2^(5/2*n))/2^(2n),#

#=2^{n+5/2*n-2n}......[because, (a^p*a^q)/a^r=a^(p+q-r)],#

#=2^(3/2*n),#

#=(2^n)^(3/2).#

#because 2^n=9, :." The Reqd. Value="9^(3/2)=(3^2)^(3/2),#

#=3^(2*3/2)=3^3=27.#