5 là số nguyên tố. Theo định lý Fermat nhỏ

\(5^{2017}-5\equiv0\) (mod 2017)

\(\Rightarrow5^{2021}=5^{2017}.5^4=\left(5^{2017}-5+5\right).5^4=\)

\(=5^4\left(5^{2017}-5\right)+5^5=5^4\left(5^{2017}-5\right)+3125=\)

\(=5^4\left(5^{2017}-5\right)+2017+1108\)

Ta có

\(5^4\left(5^{2017}-5\right)+2017⋮2017\)

\(\Rightarrow5^{2021}\equiv1108\) (mod 2017)

PTHH: \(2Mg+O_2\underrightarrow{t^o}2MgO\)

            \(4Fe+3O_2\underrightarrow{t^o}2Fe_2O_3\)

Giả sử hỗn hợp ban đầu có khối lượng 100 (g)

Gọi số mol Mg, Fe là a, b (mol)

=> 24a + 56b = 100 (1)

Theo PTHH: \(\left\{{}\begin{matrix}n_{MgO}=a\left(mol\right)\\n_{Fe_2O_3}=0,5b\left(mol\right)\end{matrix}\right.\)

m_{chất rắn sau pư} = 40a + 160.0,5b = 100.1,5

=> 40a + 80b = 150 (2)

(1)(2) => a = 1,25 (mol); b = 1,25 (mol)

=> \(\%m_{Mg}=\dfrac{1,25.24}{100}.100\%=30\%\)

\(x\left(x-8\right)=9\)

\(\Leftrightarrow x^2-8x-9=0\)

\(\Leftrightarrow x\left(x-9\right)+\left(x-9\right)=0\)

\(\Leftrightarrow\left(x-9\right)\left(x+1\right)=0\)

\(\Leftrightarrow x=9\) hoặc \(x=-1\)

\(x\left(x+14\right)=15\)

\(\Leftrightarrow x^2+14x-15=0\)

\(\Leftrightarrow x\left(x+15\right)-\left(x+15\right)=0\)

\(\Leftrightarrow\left(x+15\right)\left(x-1\right)=0\)

\(\Leftrightarrow x=-15\) hoặc \(x=1\)

\(4x\left(x-3\right)=7\)

\(\Leftrightarrow4x^2-12x-7=0\)

\(\Leftrightarrow2x\left(2x+1\right)-7\left(2x+1\right)=0\)

\(\Leftrightarrow\left(2x+1\right)\left(2x-7\right)=0\)

\(\Leftrightarrow x=-\dfrac{1}{2}\) hoặc \(x=\dfrac{7}{2}\)

\(9x\left(x-4\right)=13\)

\(\Leftrightarrow9x^2-36x-13=0\)

\(\Leftrightarrow3x\left(3x+1\right)-13\left(3x+1\right)=0\)

\(\Leftrightarrow\left(3x+1\right)\left(3x-13\right)=0\)

\(\Leftrightarrow x=-\dfrac{1}{3}\) hoặc \(x=\dfrac{13}{3}\)

\(\left(x+6\right)^2-x\left(x-3\right)=81\)

\(\Leftrightarrow x^2+12x+36-x^2+3x=81\)

\(\Leftrightarrow15x=45\)

\(\Leftrightarrow x=3\)

\(\left(2x+3\right)^2-x\left(x-3\right)=9\)

\(\Leftrightarrow4x^2+12x+9-x^2+3x=9\)

\(\Leftrightarrow3x^2+15x+9=9\)

\(\Leftrightarrow3x\left(x+5\right)=0\)

\(\Leftrightarrow x=0\) hoặc \(x=-5\)

\(\left(x-7\right)^2-x\left(x+4\right)=121\)

\(\Leftrightarrow x^2-14x+49-x^2-4x=121\)

\(\Leftrightarrow-18x=72\)

\(\Leftrightarrow x=-4\)

\(\left(3x-2\right)^2-9x\left(x-4\right)=100\)

\(\Leftrightarrow9x^2-12x+4-9x^2+36x=100\)

\(\Leftrightarrow24x=96\)

\(\Leftrightarrow x=4\)

\(\left(\dfrac{1}{2}a-\dfrac{2}{3}b\right)^3=\left(\dfrac{1}{2}a\right)^3-3.\left(\dfrac{1}{2}a\right)^2.\dfrac{2}{3}b+3.\dfrac{1}{2}a.\left(\dfrac{2}{3}b\right)^2-\left(\dfrac{2}{3}b\right)^3\)

                        \(=\dfrac{1}{8}a^3-\dfrac{1}{2}a^2b+\dfrac{2}{3}ab^2-\dfrac{8}{27}b^3\)

\(\left(\dfrac{1}{2}a-\dfrac{2}{3}b\right)^2\)

\(=\left(\dfrac{1}{2}a\right)^2-2.\left(\dfrac{1}{2}a\right).\left(\dfrac{2}{3}b\right)+\left(\dfrac{2}{3}b\right)^2\)

\(=\dfrac{1}{4}a^2-\dfrac{2}{3}ab+\dfrac{4}{9}b^2\)

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