\(A=3.\sqrt{4x^6}-3x^2\)

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

\(=3.\left|2x^3\right|-3x^2\)

\(=3.\left(-2x^3\right)-3x^2\)

\(=-6x^3-3x^2\).

\(\dfrac{5}{8}+\dfrac{5}{24}+\dfrac{5}{48}+...+\dfrac{5}{528}\)

\(=\dfrac{5}{2.4}+\dfrac{5}{4.6}+\dfrac{5}{6.8}+...+\dfrac{5}{22.24}\)

\(=\dfrac{5}{2}.\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{22.24}\right)\)

\(=\dfrac{5}{2}.\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-...-\dfrac{1}{22}+\dfrac{1}{22}-\dfrac{1}{24}\right)\)

\(=\dfrac{5}{2}.\left(\dfrac{1}{2}-\dfrac{1}{24}\right)\)

\(=\dfrac{5}{2}.\dfrac{11}{24}\)

\(=\dfrac{55}{48}\).

\(\sqrt{28-16\sqrt{3}}-\sqrt{\left(4-3\sqrt{3}\right)^2}\)

\(=\sqrt{16-16\sqrt{3}+12}-\left|4-3\sqrt{3}\right|\)

\(=\sqrt{4^2-2.4.2\sqrt{3}+\left(2\sqrt{3}\right)^2}-\left(3\sqrt{3}-4\right)\)

\(=\sqrt{\left(4-2\sqrt{3}\right)^2}-3\sqrt{3}+4\)

\(=\left|4-2\sqrt{3}\right|-3\sqrt{3}+4\)

\(=4-2\sqrt{3}-3\sqrt{3}+4\)

\(=8-5\sqrt{3}\).

\(a,\)

Biểu thức \(\sqrt{-5x-10}\) có nghĩa khi:
\(-5x-10\ge0\)

\(\Leftrightarrow5x\le-10\)

\(\Leftrightarrow x\le-2\)

Vậy \(x\le-2\).

\(b,\)

Biểu thức \(\sqrt{x^2-3x+2}\) có nghĩa khi:

\(x^2-3x+2\ge0\)

\(\Leftrightarrow x^2-x-2x+2\ge0\)

\(\Leftrightarrow x\left(x-1\right)-2\left(x-1\right)\ge0\)

\(\Leftrightarrow\left(x-2\right)\left(x-1\right)\ge0\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-2\ge0\\x-1\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}x-2\le0\\x-1\le0\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge2\\x\ge1\end{matrix}\right.\\\left\{{}\begin{matrix}x\le2\\x\le1\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x\ge2\\x\le1\end{matrix}\right.\)

Vậy \(x\ge2\) hoặc \(x\le1\).

\(c,\)

Biểu thức \(\sqrt{\dfrac{x+3}{5-x}}\) có nghĩa khi:
\(\dfrac{x+3}{5-x}\ge0\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x+3\ge0\\5-x>0\end{matrix}\right.\\\left\{{}\begin{matrix}x+3\le0\\5-x< 0\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge-3\\x< 5\end{matrix}\right.\\\left\{{}\begin{matrix}x\le-3\\x>5\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}-3\le x< 5\\5< x\le-3\left(\text{vô lí}\right)\end{matrix}\right.\)

Vậy \(-3\le x< 5\).

\(d,\)

Biểu thức \(\sqrt{-x^2+4x-4}\) có nghĩa khi:

\(-x^2+4x-4\ge0\)

\(\Leftrightarrow x^2-4x+4\le0\)

\(\Leftrightarrow\left(x-2\right)^2\le0\left(1\right)\)

Ta thấy:

\(\left(x-2\right)^2\ge0\forall x\in R\left(2\right)\)

\(\left(1\right)\left(2\right)\Rightarrow\left(x-2\right)^2=0\)

\(\Leftrightarrow x-2=0\)

\(\Leftrightarrow x=2\)

Vậy \(x=2\).

\(A,\)

Với \(x=-1;y=0,5\)

\(\Rightarrow P=\left(-1\right)^3.0,5-14.\left(-1\right)^3-6.\left(-1\right).0,5^2+0,5+2\)

\(=-0,5+14+1,5+0,5+2\)

\(=17,5\)

\(B,\)

Với \(x=0,2;y=-1,2\)

\(\Rightarrow Q=15.0,2^2.\left(-1,2\right)-5.0,2.\left(-1,2\right)^2+7.0,2.\left(-1,2\right)-21\)

\(=-0,72-1,44-1,68-21\)

\(=-24,84\).

\(A,\)

\(9x^3y^6+4x^3y^6+7x^3y^6\)

\(=\left(9+4+7\right)x^3y^6\)

\(=20x^3y^6\)

\(B,\)

\(9x^5y^6-14x^5y^6+5x^5y^6\)

\(=\left(9-14+5\right)x^5y^6\)

\(=0\).

\(45.\)

\(M=a^3+b^3+3ab\left(a^2+b^2\right)+6a^2b^2\left(a+b\right)\)

\(=\left(a+b\right)\left(a^2-ab+b^2\right)+3ab\left[\left(a^2+2ab+b^2\right)-2ab\right]+6a^2b^2\left(a+b\right)\)

\(=\left(a+b\right)\left(a^2-ab+b^2\right)+3ab\left[\left(a+b\right)^2-2ab\right]+6a^2b^2\left(a+b\right)\)

\(=a^2-ab+b^2+3ab\left(1-2ab\right)+6a^2b^2\)

\(=a^2-ab+b^2+3ab-6a^2b^2+6a^2b^2\)

\(=a^2+2ab+b^2\)

\(=\left(a+b\right)^2\)

\(=1^2\)

\(=1\).

\(S=\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2005}}\)

\(\Rightarrow2S=1+\dfrac{1}{2}+...+\dfrac{1}{2^{2004}}\)

\(\Rightarrow2S-S=\left(1+\dfrac{1}{2}+...+\dfrac{1}{2^{2004}}\right)-\left(\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{2005}}\right)\)

\(\Rightarrow S=1-\dfrac{1}{2^{2005}}\).

\(1.\)

\(a,\)

\(3x^2-6xy+3y^2\)

\(=3\left(x^2-2xy+y^2\right)\)

\(=3\left(x-y\right)^2\)

\(b,\)

\(12x^5y+24x^4y^2+12x^3y^3\)

\(=12x^3y\left(x^2+2xy+y^2\right)\)

\(=12x^3y\left(x+y\right)^2\)

\(c,\)

\(64xy-96x^2y+48x^3y-8x^4y\)

\(=8xy\left(8-12x+6x^2-x^3\right)\)

\(=8xy\left(2-x\right)^3\)

\(d,\)

\(54x^3+16y^3\)

\(=2\left(27x^3+8y^3\right)\)

\(=2\left[\left(3x\right)^3+\left(2y\right)^3\right]\)

\(=2\left(3x+2y\right)\left(9x^2-6xy+4y^2\right)\)

\(2.\)

\(a,\)

\(x^2-2xy+y^2-4\)

\(=\left(x^2-2xy+y^2\right)-4\)

\(=\left(x-y\right)^2-2^2\)

\(=\left(x-y-2\right)\left(x-y+2\right)\)

\(b,\)

\(-16x^2+8xy-y^2+49\)

\(=49-\left(16x^2-8xy+y^2\right)\)

\(=7^2-\left(4x-y\right)^2\)

\(=\left(7-4x+y\right)\left(7+4x-y\right)\)

\(3.\)

\(a,\)

\(x^6-x^4+2x^3+2x^2\)

\(=x^2\left(x^4-x^2+2x+2\right)\)

\(=x^2\left[x^2\left(x^2-1\right)+2\left(x+1\right)\right]\)

\(=x^2\left[x^2\left(x-1\right)\left(x+1\right)+2\left(x+1\right)\right]\)

\(=x^2\left(x+1\right)\left[x^2\left(x-1\right)+2\right]\)

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

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

\(=x^2\left(x+1\right)\left[x^2\left(x+1\right)-2x\left(x+1\right)+2\left(x+1\right)\right]\)

\(=x^2\left(x+1\right)\left(x+1\right)\left(x^2-2x+2\right)\)

\(=x^2\left(x+1\right)^2\left(x^2-2x+2\right)\)

\(b,\)

\(\left(x+y\right)^3-\left(x-y\right)^3\)

\(=\left(x+y-x+y\right)\left[\left(x+y\right)^2+\left(x+y\right)\left(x-y\right)+\left(x-y\right)^2\right]\)

\(=2y\left(x^2+2xy+y^2+x^2-y^2+x^2-2xy+y^2\right)\)

\(=2y\left(3x^2+y^2\right)\)