If x²+1/x² =98, find the value of x³ + 1/x³
Answers
Given : x² + 1/x² = 98
We know that, (x + 1/x)² = x² + (1/x)² + 2 × x × 1/x
[By using identity , (a + b)² = a² + b² + 2ab]
(x + 1/x)² = x² + 1/x² + 2
(x +1/x)² = 98 + 2
(x + 1/x)² = 100
(x + 1/x) = ± √10
[Taking square root on both sides]
x + 1/x = ± 10
On cubing both sides,
(x + 1/x)³ = (10)³
x³ +1/x³ + 3 x × 1/x (x + 1/x) = 1000
x³ + 1/x³ + 3 (x + 1/x) = 1000
x³ + 1/x³ + 3 (10) = 1000
x³ + 1/x³ + 30 = 1000
x³ + 1/x³ = 1000 - 30
x³ + 1/x³ = 970
Hence the value of x³ + 1/x³ is 970.
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Answer:
Step-by-step explanation:
(x + 1/x)² = x² + 1/x² + 2
(x +1/x)² = 98 + 2
(x + 1/x)² = 100
(x + 1/x) = ± √10
[Taking square root on both sides]
x + 1/x = ± 10
On cubing both sides,
(x + 1/x)³ = (10)³
x³ +1/x³ + 3 x × 1/x (x + 1/x) = 1000
x³ + 1/x³ + 3 (x + 1/x) = 1000
x³ + 1/x³ + 3 (10) = 1000
x³ + 1/x³ + 30 = 1000
x³ + 1/x³ = 1000 - 30
x³ + 1/x³ = 970