Question

How do we solve `sqrt(2^x)=4096`?

Answer 1

`x=24`

Explanation:

`sqrt(2^x)=4096`, but as `4096=2^12`

`(2^x)^(1/2)=2^12`

i.e `2^((x xx1/2))=2^12`

or `2^(x/2)=2^12`

i.e. `x/2=12`

and `x=12xx2=24`

Answer 2

`x = 24`

Explanation:

It will be to your advantage to learn some of the common powers by heart. The powers of 2 are well worth knowing.

`4096 = 2^12`

`sqrt(2^x) = 4096`

`sqrt(2^x) = 2^12" "larr` square both sides to get rid of the root

`(sqrt(2^x))^2 = (2^12)^2`

`2^x = 2^12" "larr` the bases are equal, so the indices are equal

`:. x = 24`

Answer 3

`x = 24`

Explanation:

We have: `(sqrt(2))^(x) = 4096`

Let's express all numbers in terms of `2`:

`Rightarrow (2^((1) / (2)))^(x) = 2^(12)`

Using the laws of exponents:

`Rightarrow 2^((1) / (2) x) = 2^(12)`

`Rightarrow (1) / (2) x = 12`

Finally, to solve for `x`, let's divide both sides of the equation by `(1) / (2)`:

`Rightarrow ((1) / (2) x) / ((1) / (2)) = 12 / ((1) / (2))`

`therefore x = 24`