If x + 1/x = 3, then x⁵ + 1/x⁵ is equal to?

Given equation are
3x + 2y = 12 ..............(i)
xy = 6 .........................(ii)
On squaring Eq. (i) on both sides, we will get
(3x + 2y)² = (12)²
? 9x² + 4y² + 12xy = 144
put the value of xy
? 9x²+ 4x² = 144 - 72 = 72
? 9x²+ 4x² = 72

(4 x 4 x 4 x 4 x 4 x 4)⁵ x (4 x 4 x 4)⁸ ÷ (4)³ = (64)^?
Apply the law of Fractional Exponents and Laws of Exponents
if a multiply n times a x a x a x....up to n times, then
a x a x a x a ......up to n times = aⁿ
By simplifying the equation
? (4⁶)⁵ x (4³)⁸ x 1/(4)³ = (4³)^?
? (4)³⁰ x (4)²⁴/(4)³ = (4)^{3 x ?}
? 4^{30 + 24 - 3} = 4^{3 x 7}
And comparing the exponents both the sides
? 4⁵¹ = 4^{3 x ?} ? 3 x ? = 51
? ? = 51/3 = 17

Since a * b = a + b + a / b
? 12 * 4 = 12 + 4 + 12 / 4
= 12 + 4 + 3 = 19

a² + 1 = a ? a + 1/a = 1
On squaring both sides, we get
a² + 1/a² + 2 = 1
? a² + 1/a² = -1
On cubing both sides, we get
(a² + 1/a²)³ = (-1)³
? a⁶ + 1/a⁶ + 3a² x 1/a²(a² + 1/a²) = -1
? a⁶ + 1/a⁶ + 3 x (-1) = -1
Now, a⁶ + 1/a⁶ + 1 = 3
As, a¹² + a⁶ + 1 can also be written as a⁶ + 1/a⁶ + 1
? a¹² + a⁶ + 1 = 3

We know that, a³ + b³ + c³ - 3abc
= (a + b + c) (a² + b² + c² - ab - bc - ca)
? (a + b + c) = (a³ + b³ + c³ - 3abc/a² + b² + c² - ab - bc - ca) ....(i)
Given that,
(2.247)³ + (1.730)³ + (1.023)³ - 3 x 2.247 x 1.730 x 1.023/ (2.247)² + (1.730)² + (1.023)² - (2.247 x 1.730) - (1.730 x 1.023) - (2.247 x 1.023)
= 2.247 + 1.730 x 1.023) - (2.247 x 1.023)
= (2.247 + 1.730 + 1.023) [from Eq.(i)]
= 5.000 = 5

Given that, x + 1/x = 2 ...................(i)
On squaring both sides, we get
(x + 1/x)² = 4
? x² + 1/x² + 2 = 4
? x² + 1/x² = 2 ..................................(ii)
Now, we have
(x - 1/x)² = (x² + 1/x²) - 2x X 1/x
now put the value of x² + 1/x²
(x - 1/x)² = 2 - 2 = 0 [from Eq. (ii)]
? x - 1/x = 0

x + 1/x = 6
On squaring both sides, we get
(x + 1/x)² = (6)²
? x² + 1/x² + 2 = 36
? x² + 1/x² = 34
On squaring both sides, we get
(x² + 1/x²)² = (34)²
? x⁴ + 1/x⁴ + 2 = 1156
? x⁴ + 1/x⁴ = 1154

Given in question,
x + y = 18
By using algebraic formula
? x² + y² = (x + y)² - 2xy
Put the value of x + y and xy as per given question,
? x² + y² = (18)² - 2 x 72
? x² + y² = 324 - 144
? x² + y² = 180

We know that algebraic formula,
(x + y)³ = x³ + y³ + 3xy (x + y)
put the value of x + y in given equation. [ given, x + y = 1]
1 = x³ + y³ + 3xy X 1
? x³ + y³ + 3xy = 1

p/q + q/p = (p² + q²)/pq
Apply the formula of algebra
a ²+ b² = (a + b)² - 2ab

p/q + q/p = ( (p + q)² - 2pq ) / pq
By substituting the pq and p + q values in given equation.
p/q + q/p = ( (p + q)² - 2pq ) / pq
p/q + q/p = ((10)² - 2 x 5 ) / 5
p/q + q/p = (100 - 10 )/ 5 = 90/5 = 18
p/q + q/p = 90/5 = 18
p/q + q/p = 18