If x⁴+1/x⁴ =194, find x³ + 1/x³,x² + 1/x²,+and , x + 1/x
Answers
Given : x⁴ + 1/x⁴ = 194
To find : value of x³ + 1/x³,x² + 1/x²,+and , x + 1/x
Solution :
We know that
(x² + 1/x² )² = (x²)² + (1/x²)² + 2 x² × 1/x²
(x² + 1/x² )² = x⁴ + 1/x⁴ + 2
(x² + 1/x² )² = 194 + 2
(x² + 1/x² )² = 196
(x² + 1/x² )² = 14²
x² + 1/x² = 14……….(1)
[Taking square root of both sides]
Now ,
we know that (x + 1/x)² = x² + 1/x² + 2
(x + 1/x)² = 14 + 2
[From eq 1]
(x + 1/x)² = 16
(x + 1/x)² = 4²
x + 1/x = 4 ………..(2)
On Cubing eq 2 both sides :
(x + 1/x)³ = 4³ ...........(3)
By Using Identity : (a + b)³ = a³ + b³ + 3ab(a + b)
(x)³ + (1/x)³ + 3 × x× 1/x (x + 1/x) = 64
x³ + 1/x³ - 3(x + 1/x) = 64
x³ + 1/x³ + 3(4) = 64
[From eq 3}
x³ + 1/x³ + 12 = 64
x³ + 1/x³ = 64 -12
x³ + 1/x³ = 52
Hence the value of the value of x³ + 1/x³ is 52 , x² + 1/x² is 14 and x + 1/x is 4.
HOPE THIS ANSWER WILL HELP YOU…..
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Given:
x⁴ + 1/x⁴ = 194
To find:
x³ + 1/x³ = ?
x² + 1/x² = ?
x + 1/x = ?
Solution:
Identities to be used:
(a + b)² = a² + b² + 2ab
(a + b)³ = a³ + b³ + 3ab(a + b)
(x² + 1/x²)² = x⁴ + 1/x⁴ + 2(x²)(1/x²)
=> (x² + 1/x²)² = 194 + 2
=> (x² + 1/x²)² = 196
=> x² + 1/x² = 14 __(i)
(x + 1/x)² = x² + 1/x² + 2(x)(1/x)
=> (x + 1/x)² = 14 + 2 [from (i)]
=> (x + 1/x)² = 16
=> x + 1/x = 4 __(ii)
(x + 1/x)³ = x³ + 1/x³ + 3(x)(1/x)(x + 1/x)
=> 4³ = x³ + 1/x³ + 3(4) [from (ii)]