# Resources tagged with: Powers & roots

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Broad Topics > Properties of Numbers > Powers & roots ### Rationals Between...

##### Age 14 to 16 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

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. ### Equal Temperament

##### Age 14 to 16 Challenge Level:

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. ### Roots Near 9

##### Age 14 to 16 Short Challenge Level:

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

##### Age 14 to 16 Challenge Level:

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? ### Lastly - Well

##### Age 11 to 14 Challenge Level:

What are the last two digits of 2^(2^2003)? ### Cubes Within Cubes

##### Age 7 to 14 Challenge Level:

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used? ### Guesswork

##### Age 14 to 16 Challenge Level:

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. ### Perfectly Square

##### Age 14 to 16 Challenge Level:

The sums of the squares of three related numbers is also a perfect square - can you explain why? ### Rachel's Problem

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

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? ### Sept 03

##### Age 11 to 14 Challenge Level:

What is the last digit of the number 1 / 5^903 ? ##### Age 11 to 14 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

Explore the relationships between different paper sizes. ### Double Trouble

##### Age 14 to 16 Challenge Level: ### Unusual Long Division - Square Roots Before Calculators

##### Age 14 to 16 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

Which of these pocket money systems would you rather have? ### The Root of the Problem

##### Age 14 to 18 Challenge Level:

Find the sum of this series of surds. ### Surds

##### Age 14 to 16 Challenge Level:

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay ### Enriching Experience

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

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 an Odd Fact(or)

##### Age 11 to 14 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

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. ### Negative Power

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 Challenge Level:

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... ### Deep Roots

##### Age 14 to 16 Challenge Level:

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

##### Age 11 to 14 Challenge Level:

What is the least square number which commences with six two's? ### More Magic Potting Sheds

##### Age 11 to 14 Challenge Level:

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? ### Magic Potting Sheds

##### Age 11 to 14 Challenge Level:

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? ### Power Crazy

##### Age 11 to 14 Challenge Level:

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?