### 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?

# Big Product

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

$2\times5$=$10$ and multiplying by $10$ adds a digit (a zero at the end)
$\begin{split}2^{18}\times5^{12}&=2^6\times2^{12}\times5^{12}\\ &=2^6\times\left(2^{12}\times5^{12}\right)\\ &=2^6\times\left(2\times5\right)^{12}\\ &=2^6\times10^{12}\\ &=2^6\text{ with 12 zeros afterwards}\\ &=\underbrace { 64}_{\text{2}}\underbrace{00...0}_{\text{12}}\end{split}$
So total $14$ zeros