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.

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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.

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

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?

For how many integers 𝑛 is the difference between √𝑛 and 9 is less than 1?

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

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...

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

However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

How many ways are there to count 1 - 2 - 3 in the array of triangular numbers? What happens with larger arrays? Can you predict for any size array?

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?

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

What have Fibonacci numbers got to do with Pythagorean triples?

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

Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?

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

How did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

A function pyramid is a structure where each entry in the pyramid is determined by the two entries below it. Can you figure out how the pyramid is generated?

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

The scale on a piano does something clever : the ratio (interval) between any adjacent points on the scale is equal. If you play any note, twelve points higher will be exactly an octave on.

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

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!

Find the smallest numbers a, b, and c such that: a^2 = 2b^3 = 3c^5 What can you say about other solutions to this problem?

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. . . .

Explore the relationships between different paper sizes.

In this twist on the well-known Countdown numbers game, use your knowledge of Powers and Roots to make a target.

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

Have you seen this way of doing multiplication ?

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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