Question

вычислите (√5+√2)^5​

Answer 1

$( \sqrt{5} + \sqrt{2} ) {}^{5} = ( \sqrt{5} + \sqrt{2} ) {}^{2} ( \sqrt{5} + \sqrt{2} ) {}^{3}$

$( \sqrt{5} + \sqrt{2} ) {}^{2} = 5 + 2 \sqrt{10} + 2 = 7 + 2 \sqrt{10}$

$( \sqrt{5} + \sqrt{2} ) {}^{3} = ( \sqrt{5} + \sqrt{2} ) {}^{2} ( \sqrt{5} + \sqrt{2} ) = (7 + 2 \sqrt{10} )( \sqrt{5} + \sqrt{2} ) = 7 \sqrt{5} + 7 \sqrt{2} + 2 \sqrt{50} + 2 \sqrt{20} = 7 \sqrt{5} + 7 \sqrt{2} + 10 \sqrt{2} + 4 \sqrt{5} = 11 \sqrt{5} + 17 \sqrt{2}$

$( \sqrt{5} + \sqrt{2} ) {}^{5} = (7 + 2 \sqrt{10} )(11 \sqrt{5} + 17 \sqrt{2} ) = 77 \sqrt{5} + 119 \sqrt{2} + 22 \sqrt{50} + 34 \sqrt{20} = 77 \sqrt{5} + 119 \sqrt{2} + 110 \sqrt{2} + 68 \sqrt{5} = 145 \sqrt{5} + 229 \sqrt{2}$