if x = 1-root 2 find x-1/x
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1
Answer:
x−
x
1
)
3
=8
Step-by-step explanation:
Given \: x = 1-\sqrt{2}\:---(1)Givenx=1−
2
−−−(1)
\begin{gathered}\frac{1}{x}=\frac{1}{1-\sqrt{2}}\\=\frac{1+\sqrt{2}}{(1-\sqrt{2})(1+\sqrt{2})}\end{gathered}
x
1
=
1−
2
1
=
(1−
2
)(1+
2
)
1+
2
\begin{gathered}=\frac{1+\sqrt{2}}{1^{2}-(\sqrt{2})^{2}}\\=\frac{1+\sqrt{2}}{1-2}\\=-(1+\sqrt{2})\:--(2)\end{gathered}
=
1
2
−(
2
)
2
1+
2
=
1−2
1+
2
=−(1+
2
)−−(2)
\begin{gathered}Now,\\\left(x-\frac{1}{x}\right)^{3}\\=\left(1-\sqrt{2}-[-(1+\sqrt{2})]\right)^{3}\\=\left(1-\sqrt{2}+1+\sqrt{2}\right)^{3}\\=2^{3}\\=8\end{gathered}
Now,
(x−
x
1
)
3
=(1−
2
−[−(1+
2
)])
3
=(1−
2
+1+
2
)
3
=2
3
=8
Therefore,
\left(x-\frac{1}{x}\right)^{3}=8(x−
x
1
)
3
=8
•••♪
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