Solve the equation 2 x + 2 x - 1 / 2 = 5 .

Given: 2 x + 2 x - 1 2 - 5 = 0 ⇒ 2 x + 2 x - 5 = 0 ⇒ 2 x - 5 x + 2 = 0 Let x = p then, we have 2 p 2 - 5 p + 2 = 0 ⇒ 2 p 2 - 4 p - p + 2 = 0 ⇒ 2 p p - 2 - p - 2 = 0 ⇒ 2 p - 1 p - 2 = 0 ⇒ p = 1 2 and p = 2 ⇒ x = 1 2 and x = 2 ∴ x = 1 4 or 4 .

Solve the equation 2 x + 2 x - 1 / 2 = 5 .

Given: 2 x + 2 x - 1 2 - 5 = 0 ⇒ 2 x + 2 x - 5 = 0 ⇒ 2 x - 5 x + 2 = 0 Let x = p then, we have 2 p 2 - 5 p + 2 = 0 ⇒ 2 p 2 - 4 p - p + 2 = 0 ⇒ 2 p p - 2 - p - 2 = 0 ⇒ 2 p - 1 p - 2 = 0 ⇒ p = 1 2 and p = 2 ⇒ x = 1 2 and x = 2 ∴ x = 1 4 or 4 .

Solve the equation 6 x 3 4 = 7 x 1 4 - 2 x - 1 4 .

Given: 6 x 3 4 = 7 x 1 4 - 2 x - 1 4 ⇒ 6 x 1 4 3 = 7 x 1 4 - 2 x 1 4 Putting x 1 4 = y then, we have 6 y 3 = 7 y - 2 y ⇒ 6 y 4 - 7 y 2 + 2 = 0 ⇒ 6 y 4 - 4 y 2 - 3 y 2 + 2 = 0 ⇒ 2 y 2 3 y 2 - 2 - 1 3 y 2 - 2 = 0 ⇒ 2 y 2 - 1 3 y 2 - 2 = 0 i.e., y 2 = 1 2 or 2 3 ⇒ x 1 2 = 1 2 or 2 3 ∴ x = 1 4 or 4 9 .

Solve the equation x 2 n + 6 = 5 x 1 n .

Given: x 2 n + 6 = 5 x 1 n Put x 1 n = y . Then, we have y 2 + 6 = 5 y ⇒ y 2 - 5 y + 6 = 0 ⇒ y 2 - 3 y - 2 y + 6 = 0 ⇒ ( y - 3 ) ( y - 2 ) = 0 i.e., y = 3 or 2 ∴ x 1 n = 3 or x 1 n = 2 ⇒ x = 3 n or 2 n

Solve the equation 3 x 1 2 n - x 1 n - 2 = 0 .

Given: 3 x 1 2 n - x 1 n - 2 = 0 Let y = x 1 2 n then, we have 3 y - y 2 - 2 = 0 ⇒ y 2 - 3 y + 2 = 0 ⇒ y 2 - 2 y - y + 2 = 0 ⇒ ( y - 2 ) ( y - 1 ) = 0 i.e., y = 2 , 1 ∴ x 1 2 n = 2 , 1 or x = 2 2 n , 1

Solve the equation 5 3 x + 7 x 3 = 22 2 3 .

Given: 5 3 x + 7 x 3 = 68 3 Let 3 x = y then, we have ⇒ 5 y + 7 y = 68 3 ⇒ 15 y 2 - 68 y + 21 = 0 ⇒ 15 y 2 - 63 y - 5 y + 21 = 0 ⇒ 3 y 5 y - 21 - 1 5 y - 21 = 0 ⇒ ( 3 y - 1 ) ( 5 y - 21 ) = 0 ∴ y = 1 3 , 21 5 Hence, 3 x = 1 3 , 21 5 ⇒ 3 x = 1 9 , 441 25 i.e., x = 27 , 25 147

Solve the equation x 1 - x + 1 - x x = 2 1 6 .

Given: x 1 - x + 1 - x x = 13 6 Let 1 - x x = y , then y + 1 y = 13 6 ⇒ 6 y 2 - 13 y + 6 = 0 ⇒ 6 y 2 - 9 y - 4 y + 6 = 0 ⇒ 3 y 2 y - 3 - 2 2 y - 3 = 0 ⇒ ( 3 y - 2 ) ( 2 y - 3 ) = 0 ∴ y = 2 3 , 3 2 Now, 1 - x x = y = 2 3 ⇒ 1 - x x = 4 9 ⇒ 9 - 9 x = 4 x ⇒ 13 x = 9 ∴ x = 9 13 Again, 1 - x x = y = 3 2 ⇒ 1 - x x = 9 4 ⇒ 9 x = 4 - 4 x ∴ x = 4 13 Thus, x = 4 13 , 9 13 .

Solve the equation 6 x = 5 x - 1 2 - 13 .

Given: 6 x - 5 x - 1 2 - 13 Let x = y then, we have 6 y = 5 y - 13 ⇒ 6 y 2 + 13 y - 5 = 0 ⇒ 6 y 2 + 15 y - 2 y - 5 = 0 ⇒ ( 3 y - 1 ) ( 2 y + 5 ) = 0 ∴ y = 1 3 , - 5 2 i.e., x = 1 3 , - 5 2 ∴ x = 1 9 , 25 4

Solve the equation 1 + 8 x 6 5 + 9 x 3 5 = 0 .

Given: 1 + 8 x 6 5 + 9 x 3 5 = 0 Let x 3 5 = y ⇒ x 6 5 = y 2 Then we have ⇒ 1 + 8 y 2 + 9 y = 0 ⇒ 8 y 2 + 9 y + 1 = 0 ⇒ ( 8 y + 1 ) ( y + 1 ) = 0 i.e., y = - 1 8 , - 1 ∴ x 3 5 = - 1 8 ⇒ x = - 1 8 5 3 = - 1 32 And, x 3 5 = - 1 x = ( - 1 ) 5 3 = - 1 ∴ x = - 1 32 , - 1

Solve the equation 3 2 x + 9 = 10 · 3 x .

Given: 3 2 x + 9 = 10 · 3 x Let 3 x = p then, we have p 2 - 10 p + 9 = 0 ⇒ p 2 - p - 9 p + 9 = 0 ⇒ p - 9 p - 1 = 0 i.e., p = 9 , 1 Then, 3 x = 9 = ( 3 ) 2 ∴ x = 2 And, 3 x = 1 = ( 3 ) 0 ∴ x = 0 Therefore, the required solutions are x = 2 , 0 .