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