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Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas. Rooted Via 10

How many of the numbers shown are greater than 10? Largest Expression

Which of these five algebraic expressions is largest, given $x$ is between 0 and 1?

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}
You can find more short problems, arranged by curriculum topic, in our short problems collection.