\(\sqrt{2x}.\sqrt{72x}-4x=\sqrt{2x}.\sqrt{2.36.x}-4x\)

\(=\sqrt{2x}.\sqrt{36}.\sqrt{2x}-4x=\sqrt{2x}^2.6-4x\)

\(=2x.6-4x=12x-4x=8x\)

\(\sqrt{2x}.\sqrt{72x}-4x\)

\(=\sqrt{2x.72x}-4x\)

\(=\sqrt{144x^2}-4x\)

\(=12\left|x\right|-4x\)

\(=12x-4x\left(x\ge0\right)=8x\)

\(a,\frac{6}{4+\sqrt{4-2\sqrt{3}}}=\frac{6}{4+\sqrt{\sqrt{3}^2-2\sqrt{3}+\sqrt{1}^2}}\)

\(=\frac{6}{4+\sqrt{\left(\sqrt{3}-\sqrt{1}\right)^2}}=\frac{6}{4+|\sqrt{3}-1|}=\frac{6}{3+\sqrt{3}}\)

\(=\frac{6}{\sqrt{3}\left(\sqrt{3}+1\right)}=\frac{\sqrt{36}}{\sqrt{3}\left(\sqrt{3}+1\right)}=\frac{\sqrt{3}.\sqrt{12}}{\sqrt{3}\left(\sqrt{3}+1\right)}=\frac{\sqrt{12}}{\sqrt{3}+1}\)

\(d,\frac{1}{\sqrt{7-2\sqrt{10}}}+\frac{1}{\sqrt{7+2\sqrt{10}}}\)

\(=\frac{1}{\sqrt{\sqrt{5}^2-2.\sqrt{2}.\sqrt{5}+\sqrt{2}^2}}+\frac{1}{\sqrt{\sqrt{5}^2+2.\sqrt{2}.\sqrt{5}+\sqrt{2}^2}}\)

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

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

\(=\frac{2\sqrt{5}}{\sqrt{5}^2-\sqrt{2}^2}=\frac{\sqrt{5.4}}{5-2}=\frac{\sqrt{20}}{3}\)

Ta có: \(2^{2n}=4^n\) \(\equiv4\)( mod 12) 

+) Giải thích: Vì n = 1 => \(4\equiv4\left(mod12\right)\)

Còn n > 1 ta có: \(4^{n-1}\equiv1\left(mod3\right)\Rightarrow4^n\equiv4\left(mod12\right)\)( nhân cả với 4)

Đặt: \(4^n=12k+4\)

=> \(2^{2^{2n}}=2^{12k+4}=2^{12k}.2^4\equiv1^k.16\equiv3\left(mod13\right)\)

=> \(2^{2^n}+10\equiv3+10\equiv13\equiv0\left(mod13\right)\)

=> \(2^{2^{2n}}+10⋮13\)

@Nguyễn Linh Chi : Cô ơi vậy đang mod 3 mà nhân 2 vế với 4 thì thành mod 12 ạ ?

a,\(x-\sqrt{x}-2=x-2.\frac{1}{2}.\sqrt{x}+\frac{1}{4}-\frac{9}{4}\)

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

b, \(x\sqrt{x}+8=\sqrt{x}^3+2^3=\left(\sqrt{x}+2\right)\left(x+2\sqrt{x}+4\right)\)

c, \(x-2\sqrt{x}-3=x-2.1.\sqrt{x}+1-4\)

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

d, \(x\sqrt{x}-1=\sqrt{x}^3-1^3=\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)\)

e, \(2x+3\sqrt{x}=\sqrt{x}\left(2\sqrt{x}+3\right)\)

f, \(x-7\sqrt{x}-12=\sqrt{x}^2-2.\frac{7}{2}\sqrt{x}+\frac{49}{4}-\frac{1}{4}\)

\(=\left(\sqrt{x}-\frac{7}{2}\right)^2-\left(\frac{1}{2}\right)^2=\left(\sqrt{x}-4\right)\left(\sqrt{x}-3\right)\)

\(a,\frac{2}{\sqrt{2}-1}-\frac{2}{\sqrt{2}+1}=\frac{2\left(\sqrt{2}+1\right)-2\left(\sqrt{2}-1\right)}{\left(\sqrt{2}-1\right)\left(\sqrt{2}+1\right)}\)

\(=\frac{2\sqrt{2}+2-2\sqrt{2}+2}{\sqrt{2}^2-1^2}=\frac{4}{2-1}=4\)

\(b,\sqrt{6+4\sqrt{2}}+\sqrt{6-4\sqrt{2}}\)

\(=\sqrt{4+2.2.\sqrt{2}+2}+\sqrt{4-2.2.\sqrt{2}+2}\)

\(=\sqrt{2^2+2.2.\sqrt{2}+\sqrt{2}^2}+\sqrt{2^2-2.2.\sqrt{2}+\sqrt{2}^2}\)

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

\(=|2+\sqrt{2}|+|2-\sqrt{2}|=2+2=4\)

\(c,\sqrt{9+4\sqrt{5}}+\sqrt{9-4\sqrt{5}}\)

\(=\sqrt{4+2.2.\sqrt{5}+5}+\sqrt{4-2.2.\sqrt{5}+5}\)

\(=\sqrt{2^2+2.2.\sqrt{5}+\sqrt{5}^2}+\sqrt{2^2-2.2.\sqrt{5}+\sqrt{5}^2}\)

\(=\sqrt{\left(2+\sqrt{5}\right)^2}+\sqrt{\left(2-\sqrt{5}\right)^2}\)

\(=|2+\sqrt{5}|+|2-\sqrt{5}|=2+\sqrt{5}+\sqrt{5}-2=2\sqrt{5}\)

câu d bạn cứ nhân bình thường

c, \(\sqrt{9x-9}-2\sqrt{x-1}=8\left(đk:x\ge1\right)\)

\(< =>\sqrt{9\left(x-1\right)}-2\sqrt{x-1}=8\)

\(< =>\sqrt{9}.\sqrt{x-1}-2\sqrt{x-1}=8\)

\(< =>3\sqrt{x-1}-2\sqrt{x-1}=8\)

\(< =>\sqrt{x-1}=8< =>\sqrt{x-1}=\sqrt{8}^2=\left(-\sqrt{8}\right)^2\)

\(< =>\orbr{\begin{cases}x-1=8\\x-1=-8\end{cases}< =>\orbr{\begin{cases}x=9\left(tm\right)\\x=-7\left(ktm\right)\end{cases}}}\)

d, \(\sqrt{x-1}+\sqrt{9x-9}-\sqrt{4x-4}=4\left(đk:x\ge1\right)\)

\(< =>\sqrt{x-1}+\sqrt{9\left(x-1\right)}-\sqrt{4\left(x-1\right)}=4\)

\(< =>\sqrt{x-1}+\sqrt{9}.\sqrt{x-1}-\sqrt{4}.\sqrt{x-1}=4\)

\(< =>\sqrt{x-1}+3\sqrt{x-1}-2\sqrt{x-1}=4\)

\(< =>\sqrt{x-1}\left(1+3-2\right)=4< =>2\sqrt{x-1}=4\)

\(< =>\sqrt{x-1}=\frac{4}{2}=2=\sqrt{2}^2=\left(-\sqrt{2}\right)^2\)

\(< =>\orbr{\begin{cases}x-1=2\\x-1=-2\end{cases}< =>\orbr{\begin{cases}x=3\left(tm\right)\\x=-1\left(ktm\right)\end{cases}}}\)

Đặt \(a=x^3;b=y^3;c=z^3\)thì \(\hept{\begin{cases}x,y,z>0\\xyz=1\end{cases}}\)và ta cần tìm GTLN của \(P=\frac{1}{x^3+y^3+1}+\frac{1}{y^3+z^3+1}+\frac{1}{z^3+x^3+1}\)

Áp dụng BĐT AM - GM, ta được: \(x.x.y\le\frac{x^3+x^3+y^3}{3}=\frac{2x^3+y^3}{3}\)(1) ; \(y.y.x\le\frac{y^3+y^3+x^3}{3}=\frac{2y^3+x^3}{3}\)(2)

Cộng theo vế của 2 BĐT (1) và (2), ta được: \(x^2y+xy^2\le x^3+y^3\)hay \(x^3+y^3\ge xy\left(x+y\right)\)

Kết hợp giả thiết xyz = 1 suy ra \(\frac{1}{x^3+y^3+1}\le\frac{1}{xy\left(x+y\right)+1}=\frac{1}{xy\left(x+y+z\right)}=\frac{z}{x+y+z}\)

Tương tự, ta có: \(\frac{1}{y^3+z^3+1}\le\frac{x}{x+y+z}\); \(\frac{1}{z^3+x^3+1}\le\frac{y}{x+y+z}\)

Cộng theo vế của 3 BĐT trên, ta được: \(P=\frac{1}{x^3+y^3+1}+\frac{1}{y^3+z^3+1}+\frac{1}{z^3+x^3+1}\le\frac{x+y+z}{x+y+z}=1\)

Đẳng thức xảy ra khi x = y = z = 1 hay a = b = c = 1

Không bik là đơn giản như bạn nói thật không , nhưng mik chx học tới dạng này :v

Không đơn giản thì nói làm gì.

Kẻ đường cao AH (H thuộc BC) => BH/CH=9/16

=> BH=[5:(9+16)]x9=1,8 cm => CH=5-1,8=3,2 cm

\(AH^2=BH.CH=1,8.3,2=5,76\Rightarrow AH=2,4cm\)

\(S_{ABC}=\frac{BC.AH}{2}=\frac{5.2,4}{2}=6cm^2\)