Упр.17.42 ГДЗ Мерзляк Поляков 8 класс (Алгебра)

1) v(v5-v(3-v(29-12v5) ) ) ;
2) v(3+v8) -v(3-v8) .

1) $$\sqrt{\sqrt5-\sqrt{3-\sqrt{29-12\sqrt5}}}$$

$$29-12\sqrt5=20-12\sqrt5+9=(\sqrt{20}-3)^2=(2\sqrt5-3)^2,$$
$$\sqrt{29-12\sqrt5}=2\sqrt5-3.$$

$$\sqrt{3-\sqrt{29-12\sqrt5}}=\sqrt{3-(2\sqrt5-3)}=\sqrt{6-2\sqrt5}=\sqrt{5-2\sqrt5+1}=\sqrt{(\sqrt5-1)^2}=\sqrt5-1.$$

$$\sqrt{\sqrt5-\sqrt{3-\sqrt{29-12\sqrt5}}}=\sqrt{\sqrt5-(\sqrt5-1)}=\sqrt1=1.$$

2) $$\sqrt{3+\sqrt8}-\sqrt{3-\sqrt8}$$

$$\sqrt8=2\sqrt2,$$
$$\sqrt{3+\sqrt8}-\sqrt{3-\sqrt8}=\sqrt{3+2\sqrt2}-\sqrt{3-2\sqrt2}.$$

$$3+2\sqrt2=2+2\sqrt2+1=(\sqrt2+1)^2,$$
$$3-2\sqrt2=2-2\sqrt2+1=(\sqrt2-1)^2.$$

$$\sqrt{3+2\sqrt2}-\sqrt{3-2\sqrt2}=|\sqrt2+1|-|\sqrt2-1|=\sqrt2+1-(\sqrt2-1)=2.$$

Ответ

1) $$1$$; 2) $$2$$.