Giải bài tập 3.21 trang 59

Đề bài

Rút gọn các biểu thức sau:

a) $2\sqrt{\frac{2}{3}}-4\sqrt{\frac{3}{2}};$

b) $\frac{5\sqrt{48}-3\sqrt{27}+2\sqrt{12}}{\sqrt{3}};$

c) $\frac{1}{3+2\sqrt{2}}+\frac{4\sqrt{2}-4}{2-\sqrt{2}}.$

Lời giải chi tiết

a) $2\sqrt{\frac{2}{3}}-4\sqrt{\frac{3}{2}}$$=2\frac{\sqrt{2}}{\sqrt{3}}-4\frac{\sqrt{3}}{\sqrt{2}}$$=2.\frac{\sqrt{6}}{3}-4.\frac{\sqrt{6}}{2}$$=\sqrt{6}\left(\frac{2}{3}-2\right)$$=\frac{-4\sqrt{6}}{3}.$

b) $\frac{5\sqrt{48}-3\sqrt{27}+2\sqrt{12}}{\sqrt{3}}$$=\frac{5\sqrt{16.3}-3\sqrt{9.3}+2\sqrt{4.3}}{\sqrt{3}}$$=\frac{\sqrt{3}.\left(5\sqrt{16}-3\sqrt{9}+2\sqrt{4}\right)}{\sqrt{3}}$$=5.4-3.3+2.2$$=20-9+4$$=15$

c) $\frac{1}{3+2\sqrt{2}}+\frac{4\sqrt{2}-4}{2-\sqrt{2}}$$=\frac{3-2\sqrt{2}}{\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)}+\frac{4\left(\sqrt{2}-1\right)}{\sqrt{2}\left(\sqrt{2}-1\right)}$$=\frac{3-2\sqrt{2}}{9-8}+\frac{4}{\sqrt{2}}$$=3-2\sqrt{2}+\frac{4\sqrt{2}}{2}$

$=3-2\sqrt{2}+2\sqrt{2}$$=3$