### Rationals Between

What fractions can you find between the square roots of 56 and 58?

### Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

### Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

# Power Countdown

##### Stage: 4 Challenge Level:

In the game of Power Countdown, you use a set of numbers to make a target number. The only operation you can use is raising something to a power, but you are allowed to use fractional powers - you can use a $3$ to raise a number to the power $3$ or $1\over3$.

Each number can only be used once.You don't have to use all the numbers!

Here is an example:

One way of making the target is:

Can you find any other ways of making $8$?
Are there any ways which use all the numbers?

Here is another selection.

How many ways are there to make the target number of $125$?

Below is a selection of numbers and five targets.

How many different ways can you find to make each target?

Are there any targets you can't make? How close can you get?