Question

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

Answer 1

In the above Question , the following information is given -

x⁴ + 1 / x⁴ = 194

To find -

Find the values of x² + 1 / x² and x³ + 1 / x³ respectively .

Solution -

Here ,

x⁴ + 1 / x⁴ = 194

Adding 2 on both sides -

x⁴ + 1 / x⁴ + 2 = 194 + 2

=> x⁴ + 2 ( x² )( 1 / x² ) + 1 / x⁴ = 196

=> ( x² + 1 / x² ) ² = ( 14 ) ²

=> x² + 1 / x² = 14

Now ,

x² + 1 / x² = 14

Adding 2 on both sides -

x² + 1 / x² + 2 = 14 + 2

x² + 2 ( x )( 1 / x ) + 1 / x² = 16

=> ( x + 1 / x ) ² = ( 4 ) ²

=> x + 1 / x = 4

Now , cubing both sides -

=> ( x + 1 / x ) ³ = 4 ³

=> x³ + 1 / x³ + 3 ( x + 1 / x ) = 64

=> Substituting the value of x + 1 / x -

=> x³ + 1 / x³ + 12 = 64

=> x³ + 1 / x³ = 64 - 12

=> x³ + 1 / x³ = 52 .

Answer :

The values of x² + 1 / x² and x³ + 1 / x³ are 14 and 52 respectively .

____________

Answer 2

Given:

• x⁴ + (1/x⁴) = 194

To find:

• x² + 1/x²
• x³ + 1/x³

Solution:

Given,

x⁴ + 1/x⁴ = 94

Adding 2 on Both sides

→ x⁴ + 2 + 1/x⁴ = 294 + 2

→ x⁴ + 2(x²)(1/x²) + 1/x⁴ = 196 [ °.° x² × 1/x² = 1 ]

It is in the form of

(x² + 1/x²)² = x⁴ + 2(x²)(1/x²) + 1/x⁴

→ (x² + 1/x²)² = 196

→ x² + 1/x² = √196

→ x² + 1/x² = 14

Adding 2 on both sides

→ x² + 2 + 1/x² = 14 + 2

→ x² + 2(x)(1/x) + 1/x² = 16 [°.° x × 1/x = 1]

→ (x + 1/x)² = 16

→ x + 1/x = √16

→ x + 1/x = 4 …… eq i

Cubing on both sides

→ (x + 1/x)³ = 4³

→ x³ + 1/x³ + 3{(x)(1/x)[x + 1/x]} = 64

→ x³ + 1/x³ + 3[x + 1/x] = 64

→ x³ + 1/x³ + 3(4) = 64. [°.° Eq i ]

→ x³ + 1/x³ = 64 - 12

→ x³ + 1/x³ = 56

Hence, x² + 1/x² = 4 & x³ + 1/x³ = 56