Step-by-step explanation:
We have: {x+(1/x)} = 5/2
On, squaring on both sides,we get
⇛{x+(1/x)}² = (5/2)²
Comparing the given expression with (a+b)², we get
a = x and b = 1/x
Now, using (a+b)² = a²+b²+2ab, we get
⇛x² + (1/x)² + 2(x)(1/x) = (5/2)²
Both x will cancel out because they are in multiple sign.
⇛x² + (1/x²) + 2 = {(5*5)/(2*2)}
⇛x² + (1/x²) + 2 = 25/4
Shift the number 2 from LHS to RHS, changing it's sign.
⇛x² + (1/x²) = (25/4) - 2
⇛x² + (1/x²) = (25/4 ) - (2/1)
Take the LCM of the denominator 4 & 1 is 4 is RHS.
⇛x² + (1/x²) = {(25*1 - 2*4)/4)
⇛x² + (1/x²) = {(25-8)/4}
⇛x² + (1/x²) = (17/4)
Again, squaring on both sides, we get
{x² + (1/x²)}² = (17/4)²
Comparing the given expression with (a+b)², we get
a = x² and b = (1/x²)
Now, using (a+b)² = a²+b²+2ab, we get
⇛(x²)² + (1/x²)² + 2(x²)(1/x²) = (17/4)²
⇛x⁴ + (1/x⁴) + 2(x²)(1/x²) = (17/4)²
Both x² will cancel out because they are in multiple sign.
⇛x⁴ + (1/x⁴) + 2 = {(17*17)/(4*4)}
⇛x⁴ + (1/x⁴) + 2 = (289/16)
Shift the number 2 from LHS to RHS, changing it's sign.
⇛x⁴ + (1/x⁴) = (289/16) - 2
⇛x⁴ + (1/x⁴) = (289/16) - (2/1)
Take the LCM of 16 and 1 is 16 in RHS.
⇛x⁴ + (1/x⁴) = {(289*1 - 2*16)/16}
⇛x⁴ + (1/x⁴) = {(289-32)/16}
⇛x⁴ + (1/x⁴) = (257/16)
Therefore, x⁴ + (1/x⁴) = 257/16
<u>Answer</u>: Hence, the value of x⁴ + (1/x⁴) is 257/16.
Please let me know if you have any other questions or doubt in my explanation.
