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

Correct Option: B

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 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

Correct Option: A

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 = 3 ................. (i)
On squaring both side, we will get
(x + 1/x)² = (3)²

Correct Option: A

x + 1/x = 3 ........................ (i)
On squaring both side, we will get
(x + 1/x)² = (3)²
Use the Square algebra formula (a + b)² = a² + 2ab + b²
⇒ x² + 1/x² + 2 = 9
⇒ x² + 1/x² = 7 ..................(ii)
Again squaring both sides, we will get
(x² + 1/x²)² = (7)²
Use the Square algebra formula (a + b)² = a² + 2ab + b²
⇒ x⁴ + 1/⁴ + 2 = 49
⇒ x⁴ + 1/x⁴ = 47 .....................(iii)
On cubing the equation (i) both side, we will get
(x + 1/x)³ = (3)³
Use the cube algebra formula (a + b)³ = a³ + 3a²b + 3ab² + b³
⇒ x³+ 1/x³ + 3(x + 1/x) = 27
⇒ x³ + 1/x³ + 9 = 27 [∴ (x + 1/x) = 3]
⇒ x³ + 1/x³ = 18 ...................(iv)
On multiplying Eqs. (i) and (iii) , we get
∴ (x⁴ + 1/x⁴) (x + 1/x) = 47 x 3
⇒ x⁵ + 1/x⁵ + x³ + 1/x³ = 141
⇒ x⁵ + 1/x⁵ + 18 = 141 [from Eq. (iv)]
⇒ x⁵ + 1/x⁵ = 123

Given equation are
3x + 2x = 12 ...................(i)
xy = 6 ..............................(ii)
On squaring Eq. (i) on both sides and put the value of xy and solve the equation.

Correct Option: C

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

⇒ (4⁶)⁵ x (4³)⁸ x 1/(4)³ = (4³)^?

Correct Option: A

(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