Question

What is the value of ${2^{16}}$?

Answer 1

Hint: Make use of the exponential properties and solve these types of problems. In this case make use of ${({a^m})^n} = {a^{mn}}$.

Complete Step-by-Step solution:
Here the power value =$16$
 So we know that $16 = 8 \times 2$
By the formula ${({a^m})^n} = {a^{mn}}$ we can write
$
   \Rightarrow {2^{16}} = {2^{2(8)}} \\
   \Rightarrow {2^{16}} = {4^8} \\
 $ [$\because {2^2} = 4$]
Again by using the same formula ${({a^m})^n} = {a^{mn}}$ we can write
$ \Rightarrow {2^{16}} = {4^{2(4)}}$ (where $8 = 2 \times 4$)
$
   \Rightarrow {2^{16}} = {16^4} \\
   \Rightarrow {2^{16}} = {16^{2\left( 2 \right)}} \\
   \Rightarrow {2^{16}} = {256^2} \\
   \Rightarrow {2^{16}} = 65536 \\
 $
Therefore the value ${2^{16}} = 65536$

NOTE: In this case, we have made use of the exponential law ${({a^m})^n} = {a^{mn}}$, in accordance to the problem given make use of the suitable exponential law and solve the problem.