Упр.2.149 ГДЗ Дорофеев Суворова 8 класс (Алгебра)

а) v3•v(3+v6) •v(3-v6) =3;
б) v2•v(2+v2) •v(2+v(2+v2) ) •v(2-v(2+v2) ) =2;
в) v(2+v3) •v(2+v(2+v3) ) •v(2+v(2+v(2+v3) ) ) • v(2-v(2+v(2+v3) ) ) =1.

а)

$$ \sqrt{3}\cdot \sqrt{3+\sqrt{6}}\cdot \sqrt{3-\sqrt{6}} = \sqrt{3(3+\sqrt{6})(3-\sqrt{6})} $$
$$ = \sqrt{3(9-6)}=\sqrt{3\cdot 3}=\sqrt{9}=3. $$

б)

$$ \sqrt{2}\cdot \sqrt{2+\sqrt{2}}\cdot \sqrt{2+\sqrt{2+\sqrt{2}}}\cdot \sqrt{2-\sqrt{2+\sqrt{2}}} $$
$$ = \sqrt{2}\cdot \sqrt{2+\sqrt{2}}\cdot \sqrt{\left(2+\sqrt{2+\sqrt{2}}\right)\left(2-\sqrt{2+\sqrt{2}}\right)} $$
$$ = \sqrt{2}\cdot \sqrt{2+\sqrt{2}}\cdot \sqrt{4-\left(2+\sqrt{2}\right)} $$
$$ = \sqrt{2}\cdot \sqrt{2+\sqrt{2}}\cdot \sqrt{2-\sqrt{2}} $$
$$ = \sqrt{2}\cdot \sqrt{(2+\sqrt{2})(2-\sqrt{2})} = \sqrt{2}\cdot \sqrt{4-2} $$
$$ = \sqrt{2}\cdot \sqrt{2}=2. $$

в)

$$ \sqrt{2+\sqrt{3}}\cdot \sqrt{2+\sqrt{2+\sqrt{3}}}\cdot \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}\cdot \sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}} $$
$$ = \sqrt{2+\sqrt{3}}\cdot \sqrt{2+\sqrt{2+\sqrt{3}}}\cdot \sqrt{\left(2+\sqrt{2+\sqrt{2+\sqrt{3}}}\right)\left(2-\sqrt{2+\sqrt{2+\sqrt{3}}}\right)} $$
$$ = \sqrt{2+\sqrt{3}}\cdot \sqrt{2+\sqrt{2+\sqrt{3}}}\cdot \sqrt{4-\left(2+\sqrt{2+\sqrt{3}}\right)} $$
$$ = \sqrt{2+\sqrt{3}}\cdot \sqrt{2+\sqrt{2+\sqrt{3}}}\cdot \sqrt{2-\sqrt{2+\sqrt{3}}} $$
$$ = \sqrt{2+\sqrt{3}}\cdot \sqrt{\left(2+\sqrt{2+\sqrt{3}}\right)\left(2-\sqrt{2+\sqrt{3}}\right)} $$
$$ = \sqrt{2+\sqrt{3}}\cdot \sqrt{4-\left(2+\sqrt{3}\right)} = \sqrt{2+\sqrt{3}}\cdot \sqrt{2-\sqrt{3}} $$
$$ = \sqrt{(2+\sqrt{3})(2-\sqrt{3})} = \sqrt{4-3} = \sqrt{1}=1. $$

Ответ

а) $$3$$; б) $$2$$; в) $$1$$.