Question

Solve for x:

`sqrt(27^(x + 1)) = 9^(x - 2)`

Answer 1

Given expression is `sqrt(27^(x + 1)) = 9^(x - 2)`.

We have to find the value of x in given expression.

Thus, `sqrt(27^(x + 1)) = 9^(x - 2)`

`(27^(x + 1))^(1/2) = 9^(x - 2)`  ...`[∴ root(n)(a) = a^(1/n)]`

`((3^3)^(x + 1))^(1/2) = (3^2)^(x - 2)`

`(3)^((3x + 3 1/2)) = (3)^(2x - 4)` ...[∴ (aⁿ)^m = a^nm]

`(3)^((3x + 3)/2) = (3)^(2x - 4)`  

Equating the powers with same bases.

`(3x + 3)/2 = 2x - 4`

3x + 3 = 2(2x – 4)

3x + 3 = 4x – 8

8 + 3 = 4x – 3x

11 = x

Therefore, the value of x in the given expression is 11.