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

Mar 24, 2017

$x = 24$

Explanation:

$\sqrt{{2}^{x}} = 4096$, but as $4096 = {2}^{12}$

${\left({2}^{x}\right)}^{\frac{1}{2}} = {2}^{12}$

i.e ${2}^{\left(x \times \frac{1}{2}\right)} = {2}^{12}$

or ${2}^{\frac{x}{2}} = {2}^{12}$

i.e. $\frac{x}{2} = 12$

and $x = 12 \times 2 = 24$

Mar 24, 2017

$x = 24$

Explanation:

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

$4096 = {2}^{12}$

$\sqrt{{2}^{x}} = 4096$

$\sqrt{{2}^{x}} = {2}^{12} \text{ } \leftarrow$ square both sides to get rid of the root

${\left(\sqrt{{2}^{x}}\right)}^{2} = {\left({2}^{12}\right)}^{2}$

${2}^{x} = {2}^{12} \text{ } \leftarrow$ the bases are equal, so the indices are equal

$\therefore x = 24$

Mar 24, 2017

$x = 24$

Explanation:

We have: ${\left(\sqrt{2}\right)}^{x} = 4096$

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

$R i g h t a r r o w {\left({2}^{\frac{1}{2}}\right)}^{x} = {2}^{12}$

Using the laws of exponents:

$R i g h t a r r o w {2}^{\frac{1}{2} x} = {2}^{12}$

$R i g h t a r r o w \frac{1}{2} x = 12$

Finally, to solve for $x$, let's divide both sides of the equation by $\frac{1}{2}$:

$R i g h t a r r o w \frac{\frac{1}{2} x}{\frac{1}{2}} = \frac{12}{\frac{1}{2}}$

$\therefore x = 24$