What is the value of

\(({k - {1 \over k}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})\)?

Given:

\(({k - {1 \over k}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})\)

Concept used:

(a² - b²) = (a + b)(a - b)

Calculation:

\(({k - {1 \over k}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})\)

⇒ \(\frac{({k - {1 \over k}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})({k + {1 \over k}})}{({k +{1 \over k}})}\)    [By multiplying and dividing \(k +{1 \over k}\)]

⇒ \(\frac{({k^2 - {1 \over k^2}})({k^2 + {1 \over k^2}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})}{({k +{1 \over k}})}\) 

⇒ \(\frac{({k^4 - {1 \over k^4}})({k^4 + {1 \over k^4}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})}{({k +{1 \over k}})}\) 

⇒ \(\frac{({k^8 -{1 \over k^8}})({k^8 + {1 \over k^8}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})}{({k +{1 \over k}})}\) 

⇒ \(\frac{({k^{16} - {1 \over k^{16}}})({k^{16} + {1 \over k^{16}}})({k^{32} + {1 \over k^{32}}})}{({k +{1 \over k}})}\) 

⇒ \(\frac{({k^{32} - {1 \over k^{32}}})({k^{32} + {1 \over k^{32}}})}{({k +{1 \over k}})}\) 

⇒ \(\frac{{k^{64} - {1 \over k^{64}}}}{{k +{1 \over k}}}\) 

∴ The required answer is \(\frac{{k^{64} - {1 \over k^{64}}}}{{k +{1 \over k}}}\).