The power of the sum

Answer: $n=7$

Using pairs of equal numbers

$$$\begin{array}{rl}& 2^6+2^5+2^4+2^4\\ =& 2^6+2^5+2^4\times 2\\ =& 2^6+2^5+2^{4+1}\\ =& 2^6+2^5+2^5\\ =& 2^6+2^5\times 2\\ =& 2^6+2^6\\ =& 2^7\end{array}$$$

Finding the value of the powers of 2

$2^2=4$                 $2^5=32$

$2^3=8$                 $2^6=64$

$2^4=16$               $2^7=128$

So $2^6+2^5+2^4+2^4=64+32+16+16=128=2^7$

Factorising and using index laws

Notice that all of the numbers in the sum are multiples of $2^4$, since $2^6=2^2\times2^4,2^5=2\times2^4,2^4=1\times2^4.$ So

$$\begin{array}{rl}{2}^6+2^5+2^4+2^4& =2^2\times 2^4+2\times 2^4+1\times 2^4+1\times 2^4\\ & =(2^2+2+1+1)\times 2^4\\ & =(4+2+1+1)\times 2^4\\ & =8\times 2^4\\ & =2^3\times 2^4\\ & =2^7\end{array}$$