Question : What is the value of $ \left(\mathrm{k}-\frac{1}{\mathrm{k}}\right)\left(\mathrm{k}^2+\frac{1}{\mathrm{k}^2}\right)\left(\mathrm{k}^4+\frac{1}{\mathrm{k}^4}\right)\left(\mathrm{k}^8+\frac{1}{\mathrm{k}^8}\right)\left(\mathrm{k}^{16}+\frac{1}{\mathrm{k}^{16}}\right)\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right)? $
Option 1: $\frac{\mathrm{k}^{64}-\frac{1}{\mathrm{k}^{64}}}{\mathrm{k}+\frac{1}{\mathrm{k}}}$
Option 2: $\frac{\mathrm{k}^{32}-\frac{1}{\mathrm{k}^{32}}}{\mathrm{k}-\frac{1}{\mathrm{k}}}\\$
Option 3: $\frac{\mathrm{k}^{32}-\frac{1}{\mathrm{k}^{32}}}{\mathrm{k}+\frac{1}{\mathrm{k}}}\\$
Option 4: $\frac{\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}}{\mathrm{k}+\frac{1}{\mathrm{k}}}$
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Correct Answer: $\frac{\mathrm{k}^{64}-\frac{1}{\mathrm{k}^{64}}}{\mathrm{k}+\frac{1}{\mathrm{k}}}$
Solution : $ \left(\mathrm{k}-\frac{1}{\mathrm{k}}\right)\left(\mathrm{k}^2+\frac{1}{\mathrm{k}^2}\right)\left(\mathrm{k}^4+\frac{1}{\mathrm{k}^4}\right)\left(\mathrm{k}^8+\frac{1}{\mathrm{k}^8}\right)\left(\mathrm{k}^{16}+\frac{1}{\mathrm{k}^{16}}\right)\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right) $ Multiplying and dividing by $(\mathrm{k}+\frac{1}{\mathrm{k}})$ = $ \frac{(\mathrm{k}+\frac{1}{\mathrm{k}})\left(\mathrm{k}-\frac{1}{\mathrm{k}}\right)\left(\mathrm{k}^2+\frac{1}{\mathrm{k}^2}\right)\left(\mathrm{k}^4+\frac{1}{\mathrm{k}^4}\right)\left(\mathrm{k}^8+\frac{1}{\mathrm{k}^8}\right)\left(\mathrm{k}^{16}+\frac{1}{\mathrm{k}^{16}}\right)\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right)}{(\mathrm{k}+\frac{1}{\mathrm{k}})} $ = $ \frac{(\mathrm{k}^2-\frac{1}{\mathrm{k}^2})\left(\mathrm{k}^2+\frac{1}{\mathrm{k}^2}\right)\left(\mathrm{k}^4+\frac{1}{\mathrm{k}^4}\right)\left(\mathrm{k}^8+\frac{1}{\mathrm{k}^8}\right)\left(\mathrm{k}^{16}+\frac{1}{\mathrm{k}^{16}}\right)\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right)}{(\mathrm{k}+\frac{1}{\mathrm{k}})} $ = $ \frac{(\mathrm{k}^4-\frac{1}{\mathrm{k}^4})\left(\mathrm{k}^4+\frac{1}{\mathrm{k}^4}\right)\left(\mathrm{k}^8+\frac{1}{\mathrm{k}^8}\right)\left(\mathrm{k}^{16}+\frac{1}{\mathrm{k}^{16}}\right)\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right)}{(\mathrm{k}+\frac{1}{\mathrm{k}})} $ = $ \frac{(\mathrm{k}^8-\frac{1}{\mathrm{k}^8})\left(\mathrm{k}^8+\frac{1}{\mathrm{k}^8}\right)\left(\mathrm{k}^{16}+\frac{1}{\mathrm{k}^{16}}\right)\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right)}{(\mathrm{k}+\frac{1}{\mathrm{k}})} $ = $ \frac{(\mathrm{k}^{16}-\frac{1}{\mathrm{k}^{16}})\left(\mathrm{k}^{16}+\frac{1}{\mathrm{k}^{16}}\right)\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right)}{(\mathrm{k}+\frac{1}{\mathrm{k}})} $ = $\frac{(\mathrm{k}^{32}-\frac{1}{\mathrm{k}^{32}})\left(\mathrm{k}^{32}+\frac{1}{\mathrm{k}^{32}}\right)}{(\mathrm{k}+\frac{1}{\mathrm{k}})} $ = $\frac{(\mathrm{k}^{64}-\frac{1}{\mathrm{k}^{64}})}{(\mathrm{k}+\frac{1}{\mathrm{k}})} $ Hence, the correct answer is $\frac{(\mathrm{k}^{64}-\frac{1}{\mathrm{k}^{64}})}{(\mathrm{k}+\frac{1}{\mathrm{k}})}$.
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