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The Power of the Sum

Age 14 to 16 Short Challenge Level:

Finding the value of the powers of 2
We can find each power of $2$ as a number, and then add all of the numbers together, and see what the answer is as a power of $2.$
$2^2=4$
$2^3=2\times4=8$
$2^4=2\times8=16$
$2^5=2\times16=32$
$2^6=2\times32=64$
$2^7=2\times64=128$

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


Using pairs of equal numbers
Notice $2^4+2^4$ can also be written as $2\times2^4$ or $2^1\times2^4$, which is equal to $2^5$.
So $2^6+2^5+2^4+2^4=2^6+2^5+(2^4+2^4)=2^6+2^5+2^5$.

But now we have $2^5+2^5$, which simplifies to $2^6$ in exactly the same way: $2^5+2^5=2\times2^5=2^6$.

So $2^6+2^5+2^5=2^6+2^6$, which is equal to $2\times2^6=2^7$.


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
$$\begin{align}2^6+2^5+2^4+2^4&=2^2\times2^4+2\times2^4+1\times2^4+1\times2^4\\
&=\left(2^2+2+1+1\right)\times2^4\\
&=\left(4+2+1+1\right)\times2^4\\
&=8\times2^4\\
&=2^3\times2^4\\
&=2^7\end{align}$$
You can find more short problems, arranged by curriculum topic, in our short problems collection.