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There are **54** NRICH Mathematical resources connected to **Powers and roots**, you may find related items under Properties of numbers.

Problem
Primary curriculum
Secondary curriculum
### Fit for Photocopying

Explore the relationships between different paper sizes.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Power Countdown

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Power Mad!

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Negative Power

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Equal Temperament

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### More Magic Potting Sheds

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?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Magic Potting Sheds

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?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Perfectly Square

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Guesswork

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.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What an Odd Fact(or)

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubes Within Cubes

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?

Age 7 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Rules - OK

Can you produce convincing arguments that a selection of statements about numbers are true?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Giants

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Surds

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Root to Poly

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Em'power'ed

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?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ab Surd Ity

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Archimedes Numerical Roots

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Unusual Long Division - Square Roots Before Calculators

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plus or Minus

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagorean Fibs

What have Fibonacci numbers got to do with Pythagorean triples?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Fibonacci Fashion

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Golden Eggs

Find a connection between the shape of a special ellipse and an infinite string of nested square roots.

Age 16 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### St Ives

As I was going to St Ives, I met a man with seven wives. Every wife had seven sacks, every sack had seven cats, every cat had seven kittens. Kittens, cats, sacks and wives, how many were going to St Ives?

Age 7 to 11

Problem
Primary curriculum
Secondary curriculum
### Lost in Space

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cube Roots

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Largest Number

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Enriching Experience

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Square Pair Circles

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Archimedes and Numerical Roots

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?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Power Crazy

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Deep Roots

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Two Many

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Like Powers

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.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rachel's Problem

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Consecutive Squares

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Staircase

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

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rationals Between...

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

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rational Roots

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.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Route to Root

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.?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Absurdity Again

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Age 16 to 18

Challenge Level