If x= [(2/3)²]³×(1/3)-²×3-¹×1/6 find the reciprocal of x
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Answer:
Given that, x=
2
3−
13
∴
x
1
=
3−
13
2
[Taking reciprocal on both sides]
We know that, (a+b)
2
=a
2
+b
2
+2ab
∴ (x+
x
1
)
2
=x
2
+
x
2
1
+2
⇒x
2
+
x
2
1
=(x+
x
1
)
2
−2
Now, x+
x
1
=
2
3−
13
+
3−
13
2
=
2(3−
13
)
(3−
13
)
2
+4
⇒
2(3−
13
)
9+13−6
13
+4
=
2(3−
13
)
26−6
13
⇒
2(3−
13
)
2
13
(
13
−3)
=
2(3−
13
)
−2
13
(3−
13
)
=−
13
⇒ x+
x
1
=−
13
and hence, (x+
x
1
)
2
=(−
13
)
2
=13
∴ x
2
+
x
2
1
=(x+
x
1
)
2
−2
⇒13−2=11
Hence, if x=
2
3−
13
, then the value of x
2
+
x
2
1
is 11.
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