Question

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

Answer 1

Given: x^4 + (1/x^4) = 194

To find: The value of x² + 1/x² and x³ + 1/x³.

Solution:

• Now we have provided with x⁴ + 1/x⁴ = 94

• Now adding 2 on both sides, we get:

               x⁴ + 2 + 1/x⁴ = 294 + 2

• Simplifying it, we get:

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

• So it is in the form of square, that is:

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

               (x² + 1/x²)² = 196

• Taking square root, we get:

               x² + 1/x² = √196

               x² + 1/x² = 14

• Now again adding 2 on both sides in above equation, we get:

               x² + 2 + 1/x² = 14 + 2

• Simplifying it, we get:

               x² + 2(x)(1/x) + 1/x² = 16

               (x + 1/x)² = 16

• Taking square root, we get:

               x + 1/x = √16

               x + 1/x = 4                                 ..................(i)

• Now cubing on both sides in above term, we get:

               (x + 1/x)³ = 4³

• Expanding the bracket, we get:

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

               x³ + 1/x³ + 3[x + 1/x] = 64

• Putting the value of x + 1/x = 4, we get:

               x³ + 1/x³ + 3(4) = 64                 ..................from (i)

               x³ + 1/x³ = 64 - 12

               x³ + 1/x³ = 56

Answer:

         So the value of x² + 1/x² is 14 and x³ + 1/x³ is 56.