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The Power of the Sum

Age 14 to 16 Short Challenge Level:

Answer: $n=7$

Using pairs of equal numbers
$\begin{align}&2^6+2^5+2^4+2^4\\
=&2^6 + 2^5 + 2^4\times2\\
=&2^6 + 2^5 + 2^{4+1}\\
=&2^6 + 2^5 + 2^5\\
=&2^6 + 2^5\times2\\
=&2^6 + 2^6\\
=&2^7\end{align}$


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
$$\begin{align}2^6+2^5+2^4+2^4&=2^2\times2^4+2\times2^4+1\times2^4+1\times2^4\\
&=\left(2^2+2+1+1\right)\times2^4\\
&=\left(4+2+1+1\right)\times2^4\\
&=8\times2^4\\
&=2^3\times2^4\\
&=2^7\end{align}$$
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