fbpx
calculus online - Free exercises and solutions to help you succeed!

Powers and Roots – Simplify an expression with powers – Exercise 1653

Exercise

Simplify the expression:

\frac{2^{-n+1}\cdot 4^{n-1}+2^n}{2^{n+3}-2^{n-1}}

Final Answer

\frac{2^{-n+1}\cdot 4^{n-1}+2^n}{2^{n+3}-2^{n-1}}=\frac{1}{5}

Solution

Using Powers and Roots rules we get:

\frac{2^{-n+1}\cdot 4^{n-1}+2^n}{2^{n+3}-2^{n-1}}=

=\frac{2^{-n+1}\cdot {(2^2)}^{n-1}+2^n}{2^n\cdot 2^3-2^n\cdot 2^{-1}}=

=\frac{2^{-n+1}\cdot 2^{2n-2}+2^n}{2^n\cdot 2^3-2^n\cdot 2^{-1}}=

=\frac{2^{-n+1+2n-2}+2^n}{2^n(2^3-2^{-1})}=

=\frac{2^n\cdot 2^{-1}+2^n}{2^n(2^3-2^{-1})}=

=\frac{2^n(2^{-1}+1)}{2^n(2^3-2^{-1})}=

=\frac{2^{-1}+1}{2^3-2^{-1}}=

=\frac{\frac{1}{2}+1}{8-\frac{1}{2}}=

=\frac{\frac{3}{2}}{\frac{15}{2}}=

=\frac{3}{2}\cdot\frac{2}{15}=

=\frac{6}{30}=\frac{1}{5}

Share with Friends

Leave a Reply