Powers and roots

How much money did the Queen give away in pence as a power of 2?

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

10 must remain within easy reach...

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

Find the smallest value for which a particular sequence is greater than a googol.

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.

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

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$

Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?