# Indices and Surds

### Climbing Powers

##### Age 16 to 18 Challenge Level:

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

### The Root of the Problem

##### Age 14 to 18 Challenge Level:

Find the sum of this series of surds.

### Power Stack

##### Age 16 to 18 Short Challenge Level:

When you stack powers, how do you evaluate them?

### Quick Sum

##### Age 16 to 18 Short Challenge Level:

Is this surd sum exactly 3?

### Irrational Arithmagons

##### Age 16 to 18 Challenge Level:

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?