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Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

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Rationals Between...

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

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Root to Poly

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

Power Countdown

Age 14 to 16 Challenge Level:

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?