Math, asked by Fasi3871, 9 months ago

If x⁴+1/x⁴ =194, find x³ + 1/x³,x² + 1/x²,+and , x + 1/x

Answers

Answered by nikitasingh79

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…..

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)]

=> x³ + 1/x³ = 52

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