Question
Simplify: $\frac{16\times2^{\text{n}+1}-4\times2^{\text{n}}}{16\times2^{\text{n}+2}-2\times2^{\text{n}+2}}$
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Answer
$\frac{16\times2^{\text{n}+1}-4\times2^{\text{n}}}{16\times2^{\text{n}+2}-2\times2^{\text{n}+2}}$
$=\frac{2^4\times2^{\text{n}+1}-2^2\times2^{\text{n}}}{2^4\times2^{\text{n}+2}-2\times2^{\text{n}+2}}$
$=\frac{2^{\text{n}+1+4}-2^{\text{n}+2}}{2^{\text{n}+2+4}-2^{\text{n}+2+1}}$
$=\frac{2^{\text{n}+5}-2^{\text{n}+2}}{2^{\text{n}+6}-2^{\text{n}+3}}$
$=\frac{2^\text{n}.2^5-2^\text{n}.2^2}{2^\text{n}.2^6-2^\text{n}.2^3}$
$=\frac{2^\text{n}(2^5-2^2)}{2^\text{n}(2^6-2^3)}$
$=\frac{2^5-2^2}{2^6-2^3}$
$=\frac{32-4}{64-8}=\frac{28}{56}$
$=\frac{1}{2}$
$=\frac{2^4\times2^{\text{n}+1}-2^2\times2^{\text{n}}}{2^4\times2^{\text{n}+2}-2\times2^{\text{n}+2}}$
$=\frac{2^{\text{n}+1+4}-2^{\text{n}+2}}{2^{\text{n}+2+4}-2^{\text{n}+2+1}}$
$=\frac{2^{\text{n}+5}-2^{\text{n}+2}}{2^{\text{n}+6}-2^{\text{n}+3}}$
$=\frac{2^\text{n}.2^5-2^\text{n}.2^2}{2^\text{n}.2^6-2^\text{n}.2^3}$
$=\frac{2^\text{n}(2^5-2^2)}{2^\text{n}(2^6-2^3)}$
$=\frac{2^5-2^2}{2^6-2^3}$
$=\frac{32-4}{64-8}=\frac{28}{56}$
$=\frac{1}{2}$
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