\(\left(x^4+\dfrac{1}{x^4}\right)\left(x^3+\dfrac{1}{x^3}\right)-\left(x+\dfrac{1}{x}\right)=x^7+\dfrac{x^4}{x^3}+\dfrac{x^3}{x^4}+\dfrac{1}{x^7}-x-\dfrac{1}{x}=x^7+\dfrac{1}{x^7}+x+\dfrac{1}{x}-x-\dfrac{1}{x}\)\(=x^7+\dfrac{1}{x^7}=VT\Rightarrowđpcm\)
\(b,x+\dfrac{1}{x}=7\Rightarrow\left(x+\dfrac{1}{x}\right)^2=49\)
\(\Leftrightarrow x^2+2.x.\dfrac{1}{x}+\dfrac{1}{x^2}=49\)
\(\Leftrightarrow x^2+\dfrac{1}{x^2}=49-2=47\)
\(\left(x+\dfrac{1}{x}\right)=7\Rightarrow\left(x+\dfrac{1}{x}\right)^3=343\)
\(\Leftrightarrow x^3+3x^2\dfrac{1}{x}+3x\dfrac{1}{x^2}+\dfrac{1}{x^3}=343\)
\(\Leftrightarrow x^3+\dfrac{1}{x^3}+3x\dfrac{1}{x}\left(x+\dfrac{1}{x}\right)=343\)
\(\Leftrightarrow x^3+\dfrac{1}{x^3}+3.7=343\)
\(\Leftrightarrow x^3+\dfrac{1}{x^3}=343-21=322\)
\(\left(x^2+\dfrac{1}{x^2}\right)\left(x^3+\dfrac{1}{x^3}\right)=47.322\)
\(\Leftrightarrow x^5+\dfrac{x^2}{x^3}+\dfrac{x^3}{x^2}+\dfrac{1}{x^5}=15134\)
\(\Leftrightarrow x^5+\dfrac{1}{x^5}+x+\dfrac{1}{x}=15134\)
\(\Leftrightarrow x^5+\dfrac{1}{x^5}+7=15134\)
\(\Rightarrow x^5+\dfrac{1}{x^5}=15134-7=15127\)
a. \(\left(x^4+\dfrac{1}{x^4}\right)\left(x^3+\dfrac{1}{x^3}\right)-\left(x+\dfrac{1}{x}\right)\)
\(x^7+x+\dfrac{1}{x}+\dfrac{1}{x^7}-\left(x+\dfrac{1}{x}\right)=x^7+\dfrac{1}{x^7}\)
b. Ta có:
\(\left(x+\dfrac{1}{x}\right)^2=49\)
\(\Leftrightarrow x^2+\dfrac{1}{x^2}=49-2=47\)
\(\left(x+\dfrac{1}{x}\right)^3=343\)
\(\Leftrightarrow x^3+\dfrac{1}{x^3}+3\left(x+\dfrac{1}{x}\right)=343\)
\(\Leftrightarrow x^3+\dfrac{1}{x^3}=343-3.7=322\)
\(\Rightarrow\left(x^2+\dfrac{1}{x^2}\right)\left(x^3+\dfrac{1}{x^3}\right)=47.322=15134\)
\(\Leftrightarrow x^5+\dfrac{1}{x}+x+\dfrac{1}{x^5}=15134\)
\(\Leftrightarrow x^5+\dfrac{1}{x^5}=15134-7=15127\)
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