### 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?

# Power up to 1000

##### Age 14 to 16 ShortChallenge Level

Answer: $6$

Index laws
\begin{align}1000&=10^3\\ &=\left(2\times5\right)^3\\ &=2^3\times5^3\end{align}

So $x=3$ and $y=3$, so $x+y=6$.

Calculating directly
\begin{align}1000&=10^3\\ &=10\times10\times10\\ &=\left(2\times5\right)\times\left(2\times5\right)\times\left(2\times5\right)\\ &=2\times2\times2\times5\times5\times5\\ &=2^3\times5^3\end{align}

So $x=3$ and $y=3$, so $x+y=6$.

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