If x = 3+√8 ,find the value of x3+1/x3
Answers
Step-by-step explanation:
Hey there !!
Given :-
\bf x = 3 - \sqrt{8} .x=3−
8
.
To find :-
\bf {x}^{3} + \frac{1}{ {x}^{3} } .x
3
+
x
3
1
.
▶ Solution :-
\begin{lgathered}x = 3 - \sqrt{8} . \\ \\ \frac{1}{x} = \frac{1}{3 - \sqrt{8} } . \\ \\ x + \frac{1}{x} = 3 - \sqrt{8} + \frac{1}{3 - \sqrt{8} } . \\ \\ = \frac{ {(3 - \sqrt{8} )}^{2} + 1 }{3 - \sqrt{8} } . \\ \\ = \frac{ {3}^{2} + { (\sqrt{8}) }^{2} - 2 \times 3 \times \sqrt{8} + 1}{3 - \sqrt{8} } . \\ \\ = \frac{9 + 8 - 6 \sqrt{8} + 1}{3 - \sqrt{8} } . \\ \\ = \frac{18 - 6 \sqrt{8} }{3 - \sqrt{8} } \times \frac{3 + \sqrt{8} }{3 + \sqrt{8} } . \\ \\ = \frac{(18 - 6 \sqrt{8} )(3 + \sqrt{8} )}{ {3}^{2} - {( \sqrt{8} )}^{2} } . \\ \\ = \frac{54 + \cancel{18 \sqrt{8}} - \cancel{18 \sqrt{8} }- 48 }{9 - 8} . \\ \\ = 6. \\ \\ \\ \therefore x + \frac{1}{x} = 6. \\ (cubic \: both \: side) \\ \\ = > {(x + \frac{1}{x} )}^{3} = {6}^{3} . \\ \\ = > {x}^{3} + \frac{1}{ {x}^{3} } + 3 \times x \times \frac{1}{x} (x + \frac{1}{x} ) = 216. \\ \\ = >{x}^{3} + \frac{1}{ {x}^{3} } + 3 \times 6 = 216. \\ \\ = > {x}^{3} + \frac{1}{ {x}^{3} } + 18 = 216. \\ \\ = > {x}^{3} + \frac{1}{ {x}^{3} } = 216 - 18. \\ \\ \large \boxed{ \boxed{ \bf \therefore {x}^{3} + \frac{1}{ {x}^{3} } = 198.}}\end{lgathered}
x=3−
8
.
x
1
=
3−
8
1
.
x+
x
1
=3−
8
+
3−
8
1
.
=
3−
8
(3−
8
)
2
+1
.
=
3−
8
3
2
+(
8
)
2
−2×3×
8
+1
.
=
3−
8
9+8−6
8
+1
.
=
3−
8
18−6
8
×
3+
8
3+
8
.
=
3
2
−(
8
)
2
(18−6
8
)(3+
8
)
.
=
9−8
54+
18
8
−
18
8
−48
.
=6.
∴x+
x
1
=6.
(cubicbothside)
=>(x+
x
1
)
3
=6
3
.
=>x
3
+
x
3
1
+3×x×
x
1
(x+
x
1
)=216.
=>x
3
+
x
3
1
+3×6=216.
=>x
3
+
x
3
1
+18=216.
=>x
3
+
x
3
1
=216−18.
∴x
3
+
x
3
1
=198.
✔✔ Hence, it is solved ✅✅.
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