Упр.17.45 ГДЗ Мерзляк Поляков 8 класс (Алгебра)

1) v(a+2v(a-1)) ;
2) v(a+1+4v(a-3)) ;
3) v((x+4)/4+vx) ;
4) (v(x+2v(x-3)-2)-1)/v(x-3);
5) v(2x-2v(x^2-1)) ,если x?1;
6) v(x^2+2+2v(x^2+1)) -v(x^2+2-2v(x^2+1)) ;
7) v(2a+3-2v(a^2+3a+2)) +v(a+1).

1. $$\sqrt{a+2\sqrt{a-1}}=\sqrt{(a-1)+2\sqrt{a-1}+1}=\sqrt{\left(\sqrt{a-1}+1\right)^2}=\sqrt{a-1}+1.$$

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

3. $$\sqrt{\frac{x+4}{4}+\sqrt{x}}=\sqrt{\frac{x}{4}+\frac{2\sqrt{x}}{2}+1}=\sqrt{\left(\frac{\sqrt{x}}{2}+1\right)^2}=\frac{\sqrt{x}}{2}+1.$$

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

5. $$\sqrt{2x-2\sqrt{x^2-1}}, \quad x\ne 1.$$

   Так как $$x^2-1=(x-1)(x+1),$$ то
   $$2x-2\sqrt{x^2-1}=(x+1)-2\sqrt{x^2-1}+(x-1)=\left(\sqrt{x+1}-\sqrt{x-1}\right)^2.$$
   Тогда
   $$\sqrt{2x-2\sqrt{x^2-1}}=\sqrt{x+1}-\sqrt{x-1}.$$

6. $$\sqrt{x^2+2+2\sqrt{x^2+1}}-\sqrt{x^2+2-2\sqrt{x^2+1}}$$
   $$=\sqrt{(x^2+1)+2\sqrt{x^2+1}+1}-\sqrt{(x^2+1)-2\sqrt{x^2+1}+1}$$
   $$=\sqrt{\left(\sqrt{x^2+1}+1\right)^2}-\sqrt{\left(\sqrt{x^2+1}-1\right)^2}$$
   $$=\sqrt{x^2+1}+1-\left(\sqrt{x^2+1}-1\right)=2.$$

7. $$\sqrt{2a+3-2\sqrt{a^2+3a+2}}+\sqrt{a+1}$$
   $$=\sqrt{2a+3-2\sqrt{(a+2)(a+1)}}+\sqrt{a+1}$$
   $$=\sqrt{(a+2)-2\sqrt{(a+2)(a+1)}+(a+1)}+\sqrt{a+1}$$
   $$=\sqrt{\left(\sqrt{a+2}-\sqrt{a+1}\right)^2}+\sqrt{a+1}$$
   $$=\sqrt{a+2}-\sqrt{a+1}+\sqrt{a+1}=\sqrt{a+2}.$$

Ответ

1) $$\sqrt{a-1}+1$$; 2) $$\sqrt{a-3}+2$$; 3) $$\frac{\sqrt{x}}{2}+1$$; 4) $$1$$; 5) $$\sqrt{x+1}-\sqrt{x-1}$$; 6) $$2$$; 7) $$\sqrt{a+2}$$.