Math, asked by mohinshaikh900, 9 months ago

If x⁴+1/x⁴ =119, find the value of x³ - 1/x³

Answers

Answered by nikitasingh79
9

Given : x⁴ + 1/x⁴ = 119

 

To find : value of  x³ - 1/x³

 

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 = 119 + 2

(x² + 1/x² )² = 121

(x² + 1/x² )² = 11²

x² + 1/x²  = 11……….(1)

[Taking square root of both sides]

Now ,

We know that   (x - 1/x)² = x² + 1/x² - 2

(x - 1/x)² = 11 - 2

[From eq 1]

(x - 1/x)² = 9

(x - 1/x)² = 3²

x - 1/x = 3 ………..(2)

On Cubing eq 2 both sides :

(x - 1/x)³  = 3³  

By Using Identity : (a - b)³  = a³ - b³ - 3ab(a - b)

(x)³ - (1/x)³ - 3 × x× 1/x (x - 1/x) = 27

x³ - 1/x³ - 3(x - 1/x) = 27

x³ - 1/x³ - 3(3) = 27

x³ - 1/x³ -  9 =  27

x³ - 1/x³ =  27 + 9

x³ - 1/x³ =  36

Hence the value of the value of x³ - 1/x³ is  36.

 HOPE THIS ANSWER WILL HELP YOU…..

 

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Answered by Anonymous
4

Given:

x⁴ + 1/x⁴ = 119

To find:

x³ + 1/x³ = ?

Solution:

Identities to be used:

(a + b)² = a² + b² + 2ab

(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 = 119 + 2

=> (x² + 1/x²)² = 121

=> x² + 1/x² = 11 __(i)

(x - 1/x)² = x² + 1/x² - 2(x)(1/x)

=> (x - 1/x)² = 11 - 2 [from (i)]

=> (x - 1/x)² = 9

=> x - 1/x = 3 __(ii)

(x - 1/x)³ = x³ - 1/x³ - 3(x)(1/x)(x - 1/x)

=> 3³ = x³ - 1/x³ - 3(3) [from (ii)]

=> x³ + 1/x³ = 36

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