\(3^{2^{2003}}=9^{2003}\)

Dùq mod nha ^^

9^10 = 401 (mod 100)

9^ 30 = 401 ^ 3 = 201 (mod 100)

9^120 = 201 ^ 4 = 801 ( mod 100)

9^ 360 = 801^ 3 = 401 (mod 100)

9^1080 = 401^3 = 201 (mod 100)

9^ 1800 = 9^1080. 9^ 360. 9^ 360 = 201 . 401. 401= 001 (mod 100)

9^1920 = 9^ 1800. 9^120 = 001. 801 = 801 (mod 100)

9^1980 = 9^1920. 9^ 30 . 9^ 30 = 801. 201 . 201 = 201 (mod 100)

9^2000 = 9^1980. 9^10. 9^10 = 401. 401. 201 = 001 (mod 100)

9^2003 = 9^2000. 9^ 3 = 001 . 729 = 729 (mod 100)

= là 3 dấu gạch ngang nha bạn ^^3 chữ số tận cùng là 729

9..................**** 

\(2\sqrt{x}\ge\sqrt{10}\Leftrightarrow\left(2\sqrt{x}\right)^2\ge\left(\sqrt{10}\right)^2\Leftrightarrow4x\ge10\Leftrightarrow x\ge\frac{5}{2}\)

ĐKXĐ:

\(6-2x\ne0\)

\(x\ne3\)

\(A=\frac{\sqrt{x^2-6x+9}}{6-2x}=\frac{\sqrt{\left(x-3\right)^2}}{2.\left(3-x\right)}\)

\(=\frac{\left|x-3\right|}{2.\left(3-x\right)}\)

Với \(x-3\ge0\)thì:

\(A=\frac{x-3}{2.\left(3-x\right)}=\frac{-\left(3-x\right)}{2.\left(3-x\right)}=\frac{-1}{2}\)

Với \(x-3\le0\)thì :

\(A=\frac{3-x}{2.\left(3-x\right)}=\frac{1}{2}\)

<=> 2.(x² + 2x +1) + 3.y² = 21

<=> 2.(x+1)² + 3. y² = 21

Vì 3y²; 21 đều chia hết cho 3 nên 2.(x +1)² chia hết cho 3 . hơn nữa 2. (x +1)² \(\le\) 21 và (x+1)² là số chính phương

=> (x+1)² =0 hoặc  9 

+) x + 1 = 0 => x = -1 => y ² = 7 => loại

+) (x+1)² = 9 => y² = 1

=> x+ 1 = 3 hoặc x+ 1=- 3 => x = 2 hoặc x = -4

y² = 1 => y = 1 hoặc y = -1

Vậy....

a/ \(\frac{\sqrt{ab}\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{ab}}.\left(\sqrt{a}-\sqrt{b}\right)=a-b\)

b/ \(=\left(1+\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}\right).\left(1-\frac{\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}\right)=\left(1+\sqrt{a}\right)\left(1-\sqrt{a}\right)=1-a\)

ĐKXĐ:

\(x-1\ne0\text{ và }x\ge0\)

\(x\ne1\text{ và }x\ge0\)

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

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

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

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

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

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

a)

\(P=\frac{2\sqrt{x}-1}{\sqrt{x}+1}\)

\(B=\frac{\sqrt{3}+\sqrt{11+6\sqrt{2}}-\sqrt{5+2\sqrt{6}}}{\sqrt{2}+\sqrt{6+2\sqrt{5}}-\sqrt{7+2\sqrt{10}}}\)

\(=\frac{\sqrt{3}+\sqrt{9+2.3\sqrt{2}+2}-\sqrt{3+2\sqrt{3}\sqrt{2}+2}}{\sqrt{2}+\sqrt{5+2\sqrt{5}.1+1}-\sqrt{5+2\sqrt{5}\sqrt{2}+2}}\)

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

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

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

ac*ad=ab^2