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What fractions can you find between the square roots of 65 and 67?

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

Age 14 to 16
Challenge Level

Power Countdown printable sheet

In the game of Power Countdown, you use a set of numbers to make a target number, but unlike the usual Countdown game where you can use $+, -, \times$ or $\div$, the only operations you can use are raising a number to a power, taking the reciprocal of a number, or finding the product of two numbers.

Each number can only be used once. You don't have to use all the numbers. There is often more than one way of making a particular target, so see how many different ways you can find.

Watch the video to see some examples.

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

Here is another selection.

Countdown: 2,4,5,25,27,81, target 125
How many ways are there to make the target number of $125$?

Below is a selection of numbers and five targets.
Numbers 2,5,16,243,343,512

Targets: 49,89,1024,216,64
How many different ways can you find to make each target?

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