Ta có :

A=\(\sqrt{12+6\sqrt{3}}+\sqrt{12-6\sqrt{3}}\)

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

   =\(\sqrt{3^2+2.3.\sqrt{3}+\left(\sqrt{3}\right)^2}-\sqrt{3^2-2.3\sqrt{3}+\left(\sqrt{3}\right)^2}\)

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

   =\(3+\sqrt{3}+3-\sqrt{3}=6\)

Vậy A =6

mình ghi nhầm pn ơi.. bài 2 là \(\left(3-\sqrt{2}\right)\cdot\sqrt{11+6\sqrt{6}}\)

\(\sqrt{12-6\sqrt{3}}=\sqrt{9-6\sqrt{3}+3}=\sqrt{3^2-2.3.\sqrt{3}+\left(\sqrt{3}\right)^2}=\sqrt{\left(3-\sqrt{3}\right)^2}\)

\(=\left|3-\sqrt{3}\right|=3-\sqrt{3}\)

\(\sqrt{19+8\sqrt{3}}=\sqrt{16+8\sqrt{3}+3}=\sqrt{4^2+2.4.\sqrt{3}+\left(\sqrt{3}\right)^2}=\sqrt{\left(4+\sqrt{3}\right)^2}\)

\(=\left|4+\sqrt{3}\right|=4+\sqrt{3}\)

\(\sqrt{14-6\sqrt{5}}=\sqrt{9-6\sqrt{5}+5}=\sqrt{3^2-2.3.\sqrt{5}+\left(\sqrt{5}\right)^2}=\sqrt{\left(3-\sqrt{5}\right)^2}\)

\(=\left|3-\sqrt{5}\right|=3-\sqrt{5}\)

\(\sqrt{12-6\sqrt{3}}=\sqrt{3^2-2.3.\sqrt{3}+\left(\sqrt{3}\right)^2}=\sqrt{\left(3-\sqrt{3}\right)^2}=\left|3-\sqrt{3}\right|=3-\sqrt{3}\)

\(\sqrt{19+8\sqrt{3}}=\sqrt{4^2+2.4.\sqrt{3}+\left(\sqrt{3}\right)^2}=\sqrt{\left(4+\sqrt{3}\right)^2}=\left|4+\sqrt{3}\right|=4+\sqrt{3}\)

\(\sqrt{14-6\sqrt{5}}=\sqrt{3^2-2.3.\sqrt{5}+\left(\sqrt{5}\right)^2}=\sqrt{\left(3-\sqrt{5}\right)^2}=\left|3-\sqrt{5}\right|=3-\sqrt{5}\)

học dốt quá

Cho sửa phần mẫu số của câu trên thành \(\sqrt{6}+\sqrt{2}\)

\(\frac{2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}\)

\(=\frac{2\sqrt{3+\sqrt{5-\sqrt{\left(2\sqrt{3}+1\right)^2}}}}{\sqrt{6}+\sqrt{2}}\)

\(=\frac{2\sqrt{3+\sqrt{5-|2\sqrt{3}+1|}}}{\sqrt{6}+\sqrt{2}}\)

\(=\frac{2\sqrt{3+\sqrt{4+2\sqrt{3}}}}{\sqrt{6}+\sqrt{2}}\)

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

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

\(=\frac{2\sqrt{2+\sqrt{3}}}{\sqrt{6}+\sqrt{2}}\)

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

\(=\frac{\sqrt{\left(\sqrt{3}+1\right)^2}}{\sqrt{3}+1}\)

\(=\frac{\sqrt{3}+1}{\sqrt{3}+1}=1\)

Sửa đề: \(19 + 3x + 4\sqrt{-x^2 - x + 6} = 10\sqrt{2 - x} + 10\sqrt{x + 3}\) (1)

ĐKXĐ: -3<=x<=2

Đặt \(u=\sqrt{2-x};v=\sqrt{x+3}\) (Điều kiện: u>0; v>0)

\(u^2+v^2=2-x+3+x=5\)

\(uv=\sqrt{\left(2-x\right)\left(x+3\right)}=\sqrt{2x+6-x^2-3x}=\sqrt{-x^2-x+6}\)

\(v^2=x+3\)

=>\(x=v^2-3\)

=>\(3x=3v^2-9\)

(1) sẽ trở thành: \(19 + (3v^2 - 9) + 4uv = 10u + 10v\)

=>\(10+3v^2+4uv-10u-10v=0\)

=>\(u(4v-10)=-3v^2+10v-10=-3\left(v^2-\frac{10}{3}v+\frac{10}{3}\right)=-3\left(v^2-2\cdot v\cdot\frac53+\frac{25}{9}+\frac59\right)=-3\left(v-\frac53\right)^2-\frac53<0\) ∀v

=>u(4v-10)<0

=>4v-10<0

=>4v<10

=>v<5/2

=>\(u = \frac{3v^2 - 10v + 10}{10 - 4v}\)

=>\(5 - v^2 = \left( \frac{3v^2 - 10v + 10}{10 - 4v} \right)^2\)

=>\((5 - v^2)(10 - 4v)^2 = (3v^2 - 10v + 10)^2\)

=>\((5 - v^2)(16v^2 - 80v + 100) = 9v^4 - 60v^3 + 160v^2 - 200v + 100\)

=>\(-16v^4 + 80v^3 - 20v^2 - 400v + 500 = 9v^4 - 60v^3 + 160v^2 - 200v + 100\)

=>\(25v^4 - 140v^3 + 180v^2 + 200v - 400 = 0\)

=>\(5v^4 - 28v^3 + 36v^2 + 40v - 80 = 0\)

=>\((v - 2)(5v^3 - 18v^2 + 40) = 0\)

=>v-2=0

=>v=2

=>x+3=4

=>x=1(nhận)

\(\sqrt{29-4\sqrt{7}}=\sqrt{\left(2\sqrt{7}\right)^2-2.2\sqrt{7}.1+1^2}=\sqrt{\left(2\sqrt{7}-1\right)^2}=\left|2\sqrt{7}-1\right|\)

\(=2\sqrt{7}-1\)

\(\sqrt{19+6\sqrt{2}}=\sqrt{\left(3\sqrt{2}\right)^2+2.3\sqrt{2}.1+1^2}=\sqrt{\left(3\sqrt{2}+1\right)^2}=\left|3\sqrt{2}+1\right|\)

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

\(\sqrt{28-6\sqrt{3}}=\sqrt{\left(3\sqrt{3}\right)^2-2.3\sqrt{3}.1+1^2}=\sqrt{\left(3\sqrt{3}-1\right)^2}=\left|3\sqrt{3}-1\right|\)

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

\(\sqrt{46-6\sqrt{5}}=\sqrt{\left(3\sqrt{5}\right)^2-2.3\sqrt{5}.1+1^2}=\sqrt{\left(3\sqrt{5}-1\right)^2}=\left|3\sqrt{5}-1\right|\)

\(=3\sqrt{5}-1\)

\(\sqrt{49+8\sqrt{3}}=\sqrt{\left(4\sqrt{3}\right)^2+2.4\sqrt{3}.1+1^2}=\sqrt{\left(4\sqrt{3}+1\right)^2}=\left|4\sqrt{3}+1\right|\)

\(=4\sqrt{3}+1\)

\(\sqrt{32-8\sqrt{7}}=\sqrt{\left(2\sqrt{7}\right)^2-2.2\sqrt{7}.2+2^2}=\sqrt{\left(2\sqrt{7}-2\right)^2}=\left|2\sqrt{7}-2\right|\)

\(=2\sqrt{7}-2\)

\(\sqrt{29-4\sqrt{7}}=2\sqrt{7}-1\)

\(\sqrt{19+6\sqrt{2}}=3\sqrt{2}+1\)

\(\sqrt{28-6\sqrt{3}}=3\sqrt{3}-1\)

\(\sqrt{46-6\sqrt{5}}=3\sqrt{5}-1\)

\(\sqrt{49+8\sqrt{3}}=4\sqrt{3}+1\)

\(\sqrt{32-8\sqrt{7}}=2\sqrt{7}-2\)