### Big Powers

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 $x$

##### Age 14 to 16 Short Challenge Level:

Answer: $x=3$

Using numbers
$x-1$ and $x+1$ are $2$ apart
Powers of $2$ (two columns so that numbers above each other are $2$ powers apart):
$2$        $4$
$8$       $16$    $4$ and $16$ are $12$ apart
$32$      $64$    numbers are getting too far apart

$16-4=12$
$2^4-2^2=12$
$x=3$

Using index laws
$2^{x+1}=2^x\times2^1=2^x\times2$ and $2^{x-1}=2^x\times2^{-1}=2^x\times\frac12$.

Substitute then factorise: \begin{align} 2^{x+1}-2^{x-1} =&12\\ \Rightarrow2^x\times2-2^x\times\tfrac12=&12\\ \Rightarrow 2^x\left(2-\tfrac12\right)=&12\\ \Rightarrow 2^x\times\tfrac32=&12\\ \Rightarrow 2^x\times3=&12\times2\\ \Rightarrow 2^x=&24\div3\\ \Rightarrow 2^x=&8\end{align} So $x=3$, since $2^3=8$.
You can find more short problems, arranged by curriculum topic, in our short problems collection.