Question

Aratati ca fractia

2¹×2²×2³×......×2¹⁰⁰/(32¹⁸)⁵⁶ este supraunitara​

Answer 1

$\frac{{2}^{1} \times {2}^{2} \times ... \times {2}^{100} }{(32 {}^{18} ) {}^{56} } \\ \\ = \frac{2 {}^{1 + 2 + ... + 100} }{32 {}^{18 \times 56} } = \frac{2 {}^{ \frac{100 \times 101}{2} } }{32 {}^{1008} } = \frac{2 {}^{50 \times 101} }{(2 {}^{5} ) {}^{1008} } \\ \\ = \frac{ {2}^{5050} }{2 {}^{5 \times 1008} } = \frac{ {2}^{5050} }{2 {}^{5040} } \implies \: {2}^{5050} > {2}^{5040} \\ \\ \iff \:\text{ este \: supraunitara}.$