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There are **37** NRICH Mathematical resources connected to **Indices**, you may find related items under Algebraic expressions, equations and formulae.

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
### Tens

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?

Age 16 to 18

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
### Sums of Squares

Can you prove that twice the sum of two squares always gives the sum of two squares?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Power Quady

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

Age 16 to 18

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
### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Mega Quadratic Equations

What do you get when you raise a quadratic to the power of a quadratic?

Age 14 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Card Shuffle

This article for students and teachers tries to think about how long would it take someone to create every possible shuffle of a pack of cards, with surprising results.

Age 11 to 16

Article
Primary curriculum
Secondary curriculum
### Public Key Cryptography

An introduction to coding and decoding messages and the maths behind how to secretly share information.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### The Public Key

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Multiplication Magic

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). The question asks you to explain the trick.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Powerful Factors

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras Mod 5

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.

Age 16 to 18

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

Article
Primary curriculum
Secondary curriculum
### Learn about Number Bases

We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.

Age 11 to 18

Article
Primary curriculum
Secondary curriculum
### Modulus Arithmetic and a Solution to Dirisibly Yours

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Sums of Squares and Sums of Cubes

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Really Mr. Bond

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Age 14 to 16

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
### 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
### Even So

Find some triples of whole numbers a, b and c such that a^2 + b^2 + c^2 is a multiple of 4. Is it necessarily the case that a, b and c must all be even? If so, can you explain why?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Thirty Six Exactly

The number 12 = 2^2 × 3 has 6 factors. What is the smallest natural number with exactly 36 factors?

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
### 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
### Remainder Hunt

What are the possible remainders when the 100-th power of an integer is divided by 125?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Power Up

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Big, Bigger, Biggest

Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Biggy

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

Age 16 to 18

Challenge Level