Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

What is the least square number which commences with six two's?

What is the largest number you can make using the three digits 2, 3 and 4 in any way you like, using any operations you like? You can only use each digit once.

Can you find what the last two digits of the number $4^{1999}$ are?

What can you say about the values of n that make $7^n + 3^n$ a multiple of 10? Are there other pairs of integers between 1 and 10 which have similar properties?

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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

Powers of numbers behave in surprising ways. Take a look at some of these and try to explain why they are true.

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

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...

Have you seen this way of doing multiplication ?

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?

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

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

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.

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

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

Which of these pocket money systems would you rather have?

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

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

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

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

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.

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

Explore the relationships between different paper sizes.

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