Question : What is the value of $\left(k-\frac{1}{k}\right)\left(k^2+\frac{1}{k^2}\right)\left(k^4+\frac{1}{k^4}\right)\left(k^8+\frac{1}{k^8}\right)\left(k^{16}+\frac{1}{k^{16}}\right) ?$
Option 1: $k^{64}-\frac{1}{k^{64}}$
Option 2: $\frac{k^{32}-\frac{1}{k^{32}}}{k-\frac{1}{k}}$
Option 3: $k^{32}-\frac{1}{k^{32}}$
Option 4: $\frac{k^{32}-\frac{1}{k^{32}}}{k+\frac{1}{k}}$
New: SSC CHSL tier 1 answer key 2024 out | SSC CHSL 2024 Notification PDF
Recommended: How to crack SSC CHSL | SSC CHSL exam guide
Don't Miss: Month-wise Current Affairs | Upcoming government exams
Correct Answer: $\frac{k^{32}-\frac{1}{k^{32}}}{k+\frac{1}{k}}$
Solution :
Consider, $\left(k-\frac{1}{k}\right)\left(k^2+\frac{1}{k^2}\right)\left(k^4+\frac{1}{k^4}\right)\left(k^8+\frac{1}{k^8}\right)\left(k^{16}+\frac{1}{k^{16}}\right)$
Multiplying and dividing by $(k+\frac{1}{k})$
⇒ $\left(k-\frac{1}{k}\right)\left(k^2+\frac{1}{k^2}\right)\left(k^4+\frac{1}{k^4}\right)\left(k^8+\frac{1}{k^8}\right)\left(k^{16}+\frac{1}{k^{16}}\right)=\frac{(k-\frac{1}{k})(k+\frac{1}{k})(k^2+\frac{1}{k^2})(k^4+\frac{1}{k^4})(k^8+\frac{1}{k^8})(k^{16}+\frac{1}{k^{16}})}{k+\frac{1}{k}}$
Now using, $(a+b)(a-b)=a^2 - b^2$
⇒ $\left(k-\frac{1}{k}\right)\left(k^2+\frac{1}{k^2}\right)\left(k^4+\frac{1}{k^4}\right)\left(k^8+\frac{1}{k^8}\right)\left(k^{16}+\frac{1}{k^{16}}\right)=\frac{(k^2-\frac{1}{k^2})(k^2+\frac{1}{k^2})(k^4+\frac{1}{k^4})(k^8+\frac{1}{k^8})(k^{16}+\frac{1}{k^{16}})}{k+\frac{1}{k}}$
$=\frac{(k^4-\frac{1}{k^4})(k^4+\frac{1}{k^4})(k^8+\frac{1}{k^8})(k^{16}+\frac{1}{k^{16}})}{k+\frac{1}{k}}$
$=\frac{(k^8-\frac{1}{k^8})(k^8+\frac{1}{k^8})(k^{16} + \frac{1}{k^{16}})}{k+\frac{1}{k}}$
$=\frac{(k^{16} - \frac{1}{k^{16}})(k^{16} + \frac{1}{k^{16}})}{k+\frac{1}{k}}$
$=\frac{k^{32} - \frac{1}{k^{32}}}{k+\frac{1}{k}}$
Hence, the correct answer is $\frac{k^{32} - \frac{1}{k^{32}}}{k+\frac{1}{k}}$.
Related Questions
Know More about
Staff Selection Commission Combined High ...
Result | Eligibility | Application | Admit Card | Answer Key | Preparation Tips | Cutoff
Get Updates BrochureYour Staff Selection Commission Combined Higher Secondary Level Exam brochure has been successfully mailed to your registered email id “”.