### Big, Bigger, Biggest

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

### Complex Sine

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.

### Sierpinski Triangle

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

# Power Match

### Why do this problem?

This interactivity gives a really good workout in the use of powers and logarithms. Even students who feel technically capable of manipulating powers might turn to have relatively little intuition as to how 'large' non-integer powers are.Various computational tricks might emerge as students attempt to calculate irregular powers. For example, since $2^3$ is less than 10, we know that $2^{18}$ will be less than $10^6$.

### Possible approach

This activity can be played as a game or individually. When a flag is incorrectly placed, can students understand why? This is good for rooting out errors in understanding. For example, $10^{1.5}$ is not half way between 10 and 100. Why not?

### Key questions

Do you know what the meaning of the power you are trying to place is? Can you explain in words?
Can you relate the power in question to a simpler approximation?

### Possible extension

Can students devise any computational tricks to help to evaluate powers?

### Possible support

You can initially increase the range of acceptable accuracy. Keep practising and use trial and error to hone in on the answer.