If x = 3 + 2 √2, find the value of √x -1/√x
Answers
Answer:
Hence, x
3
+
x
3
1
=198
Step-by-step explanation:
It is given that x=3−2
2
, so we first find x
3
that is (3−2
2
)
3
as follows:
(3−2
2
)
3
=(3−2
2
){(3)
2
+(2
2
)
2
−(2×3×2
2
)}{∵(a−b)
3
=(a−b)(a
2
+b
2
−2ab)}
=(3−2
2
){9+8−(12
2
)}
=(3−2
2
)(17−12
2
)
=(3×17)+[3×(−12
2
)]+[(−2
2
)×17]+[(−2
2
)×(−12
2
)]
=51−36
2
−34
2
+48
=99−70
2
Therefore, x
3
=99−70
2
Now we will find the value of x
3
+
x
3
1
as shown below:
x
3
+
x
3
1
=99−70
2
+
99−70
2
1
=99−70
2
+(
99−70
2
1
×
99+70
2
99+70
2
){Rationalizedenominator}
=99−70
2
+(
(99−70
2
)(99+70
2
)
99+70
2
)
=99−70
2
+(
(99)
2
−(70
2
)
2
99+70
2
){∵(a−b)(a+b)=a
2
−b
2
}
=99−70
2
+(
9801−9800
99+70
2
)
=99−70
2
+(
1
99+70
2
)
=99−70
2
+99+70
2
=99+99
=198
Hence, x
3
+
x
3
1
=198