Powers and roots

The English mathematician Augustus de Morgan has given his age in algebraic terms. Can you work out when he was born?

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this sequence. What do you notice? Calculate a few more terms and find the squares of the terms. Can you prove that the special property you notice about this sequence will apply to all the later terms of the sequence? Write down a formula to give an approximation to the cube root of a number and test it for the cube root of 3 and the cube root of 8. How many terms of the sequence do you have to take before you get the cube root of 8 correct to as many decimal places as your calculator will give? What happens when you try this method for fourth roots or fifth roots etc.?

Investigate the number of points with integer coordinates on circles with centres at the origin for which the square of the radius is a power of 5.

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

10 must remain within easy reach...

Find the five distinct digits N, R, I, C and H in the following nomogram

What have Fibonacci numbers got to do with Pythagorean triples?

Powers of numbers might look large, but which of these is the largest...

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

Make and prove a conjecture about the value of the product of the Fibonacci numbers $F_{n+1}F_{n-1}$.