Question

If x= 3+root 8 then find the value of xraised to the power4 +1 divided by x raised to the power 4

Answer 1

Answer:

x⁴ + 1/x⁴ = 1154

Solution:

• Given : x = 3 + √8
• To find ; x⁴ + 1/x⁴ = ?

We have ;

x = 3 + √8

Thus,

1/x = 1/(3 + √8)

Now,

Rationalising the denominator of the term in RHS , we have ;

=> 1/x = (3 - √8)/(3 + √8)(3 - √8)

=> 1/x = (3 - √8)/[ 3² - (√8)² ]

=> 1/x = (3 - √8)/(9 - 8)

=> 1/x = (3 - √8)/1

=> 1/x = 3 - √8

Now,

=> x + 1/x = 3 + √8 + 3 - √8

=> x + 1/x = 6

Now,

Squaring both sides , we get ;

=> (x + 1/x)² = 6²

=> x² + 2•x•(1/x) + (1/x)² = 36

=> x² + 2 + 1/x² = 36

=> x² + 1/x² = 36 - 2

=> x² + 1/x² = 34

Again,

Squaring both sides , we get ;

=> (x² + 1/x²)² = 34²

=> (x²)² + 2•x²•(1/x²) + (1/x²)² = 1156

=> x⁴ + 2 + 1/x⁴ = 1156

=> x⁴ + 1/x⁴ = 1156 - 2

=> x⁴ + 1/x⁴ = 1154

Hence,

x⁴ + 1/x⁴ = 1154