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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.