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\(x^2+\frac{1}{x^2}=7\Leftrightarrow x^2+2.x.\frac{1}{x}+\frac{1}{x^2}=9\)
\(\Leftrightarrow\left(x+\frac{1}{x}\right)^2=9\Leftrightarrow x+\frac{1}{x}=3\)
\(P=x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)^3-3x.\frac{1}{x}\left(x+\frac{1}{x}\right)=3^3-3.3=18\)
\(Q=\left(x^3+\frac{1}{x^3}\right)\left(x^2+\frac{1}{x^2}\right)-\left(x+\frac{1}{x}\right)=7.18-3=...\)
Ta có :
\(x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)\left(x^2-1+\frac{1}{x^2}\right)\)
\(=\left(x+\frac{1}{x}\right)\left(7-1\right)\)(vì \(x^2+\frac{1}{x^2}=7\))
\(=6\left(x+\frac{1}{x}\right)\)
Đặt \(x+\frac{1}{x}=a\)thì \(\left(x+\frac{1}{x}\right)=a^2\). Suy ra \(a^2-2=x^2+\frac{1}{x^2}\)
\(\Rightarrow a^2-2=7\)(vì \(x^2+\frac{1}{x^2}=7\))
\(\Rightarrow a^2=9\)\(\Rightarrow\left(x+\frac{1}{x}\right)^2=9\)
Vì \(x\inℝ,x>0\)nên \(x+\frac{1}{x}>0\)
\(\Rightarrow\) \(\left(x+\frac{1}{x}\right)^2=3^2\Rightarrow x+\frac{1}{x}=3\)
Do đó \(x^3+\frac{1}{x^3}=6.3=18\)
Ta có:
\(\left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=x^5+\frac{1}{x^5}+1\)
Mà \(\left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=7.18=126\)
\(\Rightarrow x^5+\frac{1}{x^5}+1=126\)
\(\Rightarrow x^5+\frac{1}{x^5}=125\)
Vậy với \(x\inℝ,x>0\)và \(x^2+\frac{1}{x^2}=7\)thì \(x^5+\frac{1}{x^5}=125\)
\(e)\) \(\left|2x-3\right|=x-1\)
Ta có :
\(\left|2x-3\right|\ge0\)\(\left(\forall x\inℚ\right)\)
Mà \(\left|2x-3\right|=x-1\)
\(\Rightarrow\)\(x-1\ge0\)
\(\Rightarrow\)\(x\ge1\)
\(\Leftrightarrow\)\(\orbr{\begin{cases}2x-3=x-1\\2x-3=1-x\end{cases}\Leftrightarrow\orbr{\begin{cases}2x-x=-1+3\\2x+x=1+3\end{cases}}}\)
\(\Leftrightarrow\)\(\orbr{\begin{cases}x=2\\3x=4\end{cases}\Leftrightarrow\orbr{\begin{cases}x=2\left(tm\right)\\x=\frac{4}{3}\left(tm\right)\end{cases}}}\)
Vậy \(x=2\) hoặc \(x=\frac{4}{3}\)
Chúc bạn học tốt ~
\(f)\) \(\left|x-5\right|-5=7\)
\(\Leftrightarrow\)\(\left|x-5\right|=12\)
\(\Leftrightarrow\)\(\orbr{\begin{cases}x-5=12\\x-5=-12\end{cases}\Leftrightarrow\orbr{\begin{cases}x=17\\x=-7\end{cases}}}\)
Vậy \(x=17\) hoặc \(x=-7\)
Chúc bạn học tốt ~
ta có \(x^2+\frac{1}{x^2}\)
=\(\left(x+\frac{1}{x}\right)^2-2x\frac{1}{x}=\left(x+\frac{1}{x}\right)^2-2\)
=> \(\left(x+\frac{1}{x}\right)^2=25.vì\)\(x>0\Rightarrow x+\frac{1}{x}>0\Rightarrow x+\frac{1}{x}=5\)
\(\left(x+\frac{1}{x}\right)^3=x^3+\frac{1}{x^3}+3x+\frac{3}{x}=x^3+\frac{1}{x^3}+15\)
\(\Rightarrow x^3+\frac{1}{x^3}=5^3+15=110\)
\(\left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=x^5+\frac{1}{x^5}+x+\frac{1}{x}=x^5+\frac{1}{x^5}+5\)
\(\Rightarrow x^5+\frac{1}{x^5}=23\cdot110-5=2525\)
Vậy...
2/ \(\frac{1}{2}x2y5z3=\left(\frac{1}{2}.2.5.3\right)xyz\)\(=15xyz\)
\(\Rightarrow\frac{1}{2}x2y5z3\)có bậc là 3
3/ \(\frac{x}{4}=\frac{9}{x}\Leftrightarrow x^2=9.4\Rightarrow x^2=36\) mà \(x>0\Rightarrow x=6\)
4/ \(\left|2x-\frac{1}{2}\right|+\frac{3}{7}=\frac{38}{7}\Rightarrow\left|2x+\frac{1}{2}\right|=\frac{35}{7}=5\Rightarrow\hept{\begin{cases}2x+\frac{1}{2}=5\Rightarrow2x=\frac{9}{2}\Rightarrow x=\frac{9}{4}\\2x+\frac{1}{2}=-5\Rightarrow2x=\frac{-11}{2}\Rightarrow x=\frac{-11}{4}\end{cases}}\)
\( {x^2} + \dfrac{1}{{{x^2}}} = 7 \Rightarrow {\left( {x + \dfrac{1}{x}} \right)^2} = 9 \Rightarrow x + \dfrac{1}{x} = 3\left( {x > 0} \right)\\ \Rightarrow \left( {x + \dfrac{1}{x}} \right)\left( {{x^2} + \dfrac{1}{{{x^2}}}} \right) = 21 \Rightarrow {x^2} + \dfrac{1}{{{x^2}}} = 18\\ \Rightarrow \left( {{x^2} + \dfrac{1}{{{x^2}}}} \right)\left( {{x^2} + \dfrac{1}{{{x^2}}}} \right) = 7.18\\ \Rightarrow {x^5} + \dfrac{1}{{{x^5}}} = 123 ]\)
(x+\(\frac{1}{x}\))2=9⇒x+\(\frac{1}{x}\)=3 ; (x2+\(\frac{1}{x^2}\))2=49⇒x4+\(\frac{1^{ }}{x^4}\)=47 và (x+\(\frac{1}{x}\))(x2+\(\frac{1}{x^2}\))=x3+\(\frac{1}{x^3}\)+x+\(\frac{1}{x}\)=21⇒x3+\(\frac{1}{x^3}\)=18
⇒(x+\(\frac{1}{x}\))(x4+\(\frac{1}{x^4}\))=141
⇒x5+\(\frac{1}{x^3}\)+x3+\(\frac{1}{x^5}\)=141
⇒x5+\(\frac{1}{x^5}\) =141-18=123