Question

[Nguyễn Ngọc Gia Hân]

Rút gọn:

a. \(\sqrt{6+\sqrt{24}+\sqrt{12}+\sqrt{8}}-\sqrt{4-2\sqrt{3}}\)

b.\(\left(\frac{15}{\sqrt{6}+1}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\right).\left(\sqrt{6}+11\right)\)

Answer 1

[Akai Haruma]

Lời giải:

a)

\(\sqrt{6+\sqrt{24}+\sqrt{12}+\sqrt{8}}-\sqrt{4-2\sqrt{3}}\)

\(=\sqrt{6+2\sqrt{6}+2\sqrt{3}+2\sqrt{2}}-\sqrt{3+1-2\sqrt{3}}\)

\(=\sqrt{(3+1+2\sqrt{3})+2+(2\sqrt{6}+2\sqrt{2})}-\sqrt{(\sqrt{3}-\sqrt{1})^2}\)

\(=\sqrt{(\sqrt{3}+1)^2+2\sqrt{2}(\sqrt{3}+1)+2}-\sqrt{(\sqrt{3}-1)^2}\)

\(=\sqrt{(\sqrt{3}+1+\sqrt{2})^2}-\sqrt{(\sqrt{3}-1)^2}\)

\(=\sqrt{3}+1+\sqrt{2}-(\sqrt{3}-1)=2+\sqrt{2}\)

b)

\(\left(\frac{15}{\sqrt{6}+1}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\right)(\sqrt{6}+11)\)

\(=\left(\frac{15(\sqrt{6}-1)}{(\sqrt{6}+1)(\sqrt{6}-1)}+\frac{4(\sqrt{6}+2)}{(\sqrt{6}-2)(\sqrt{6}+2)}-\frac{12(3+\sqrt{6})}{(3-\sqrt{6})(3+\sqrt{6})}\right)(\sqrt{6}+11)\)

\(=\left(\frac{15(\sqrt{6}-1)}{5}+\frac{4(\sqrt{6}+2)}{2}-\frac{12(3+\sqrt{6})}{3}\right)(\sqrt{6}+11)\)

\(=[3(\sqrt{6}-1)+2(\sqrt{6}+2)-4(3+\sqrt{6})](\sqrt{6}+11)\)

\(=(\sqrt{6}-11)(\sqrt{6}+11)=6-11^2=-115\)

Answer 2

[Akai Haruma]

Lời giải:

a)

\(\sqrt{6+\sqrt{24}+\sqrt{12}+\sqrt{8}}-\sqrt{4-2\sqrt{3}}\)

\(=\sqrt{6+2\sqrt{6}+2\sqrt{3}+2\sqrt{2}}-\sqrt{3+1-2\sqrt{3}}\)

\(=\sqrt{(3+1+2\sqrt{3})+2+(2\sqrt{6}+2\sqrt{2})}-\sqrt{(\sqrt{3}-\sqrt{1})^2}\)

\(=\sqrt{(\sqrt{3}+1)^2+2\sqrt{2}(\sqrt{3}+1)+2}-\sqrt{(\sqrt{3}-1)^2}\)

\(=\sqrt{(\sqrt{3}+1+\sqrt{2})^2}-\sqrt{(\sqrt{3}-1)^2}\)

\(=\sqrt{3}+1+\sqrt{2}-(\sqrt{3}-1)=2+\sqrt{2}\)

b)

\(\left(\frac{15}{\sqrt{6}+1}+\frac{4}{\sqrt{6}-2}-\frac{12}{3-\sqrt{6}}\right)(\sqrt{6}+11)\)

\(=\left(\frac{15(\sqrt{6}-1)}{(\sqrt{6}+1)(\sqrt{6}-1)}+\frac{4(\sqrt{6}+2)}{(\sqrt{6}-2)(\sqrt{6}+2)}-\frac{12(3+\sqrt{6})}{(3-\sqrt{6})(3+\sqrt{6})}\right)(\sqrt{6}+11)\)

\(=\left(\frac{15(\sqrt{6}-1)}{5}+\frac{4(\sqrt{6}+2)}{2}-\frac{12(3+\sqrt{6})}{3}\right)(\sqrt{6}+11)\)

\(=[3(\sqrt{6}-1)+2(\sqrt{6}+2)-4(3+\sqrt{6})](\sqrt{6}+11)\)

\(=(\sqrt{6}-11)(\sqrt{6}+11)=6-11^2=-115\)