# Resources tagged with: Powers & roots

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There are 41 NRICH Mathematical resources connected to Powers & roots, you may find related items under Properties of Numbers.

Broad Topics > Properties of Numbers > Powers & roots

### Fit for Photocopying

##### Age 14 to 16Challenge Level

Explore the relationships between different paper sizes.

### Negative Power

##### Age 14 to 16Challenge Level

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

### Equal Temperament

##### Age 14 to 16Challenge 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.

### Perfectly Square

##### Age 14 to 16Challenge Level

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

### Guesswork

##### Age 14 to 16Challenge 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.

### In Between

##### Age 16 to 18Challenge Level

Can you find the solution to this algebraic inequality?

### The Root of the Problem

##### Age 14 to 18Challenge Level

Find the sum of this series of surds.

### Number Rules - OK

##### Age 14 to 16Challenge Level

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

### Giants

##### Age 16 to 18Challenge Level

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

### Surds

##### Age 14 to 16Challenge 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

### Root to Poly

##### Age 14 to 16Challenge Level

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

### How Many Solutions?

##### Age 16 to 18Challenge Level

Find all the solutions to the this equation.

### Em'power'ed

##### Age 16 to 18Challenge Level

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?

### Ab Surd Ity

##### Age 16 to 18Challenge Level

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

### Roots Near 9

##### Age 14 to 16 ShortChallenge Level

For how many integers ð‘› is the difference between âˆšð‘› and 9 is less than 1?

### Function Pyramids

##### Age 16 to 18Challenge Level

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?

### Irrational Arithmagons

##### Age 16 to 18Challenge Level

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

### Googol

##### Age 16 to 18 ShortChallenge Level

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

### Archimedes Numerical Roots

##### Age 16 to 18Challenge Level

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

### Power Countdown

##### Age 14 to 16Challenge Level

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

### Unusual Long Division - Square Roots Before Calculators

##### Age 14 to 16Challenge 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?

### Bina-ring

##### Age 16 to 18Challenge Level

Investigate powers of numbers of the form (1 + sqrt 2).

### Plus or Minus

##### Age 16 to 18Challenge Level

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

### Pythagorean Fibs

##### Age 16 to 18Challenge Level

What have Fibonacci numbers got to do with Pythagorean triples?

### Fibonacci Fashion

##### Age 16 to 18Challenge Level

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

### Golden Eggs

##### Age 16 to 18Challenge Level

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

### Lost in Space

##### Age 14 to 16Challenge 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?

### Cube Roots

##### Age 16 to 18Challenge Level

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

### Enriching Experience

##### Age 14 to 16Challenge Level

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

### Square Pair Circles

##### Age 16 to 18Challenge Level

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.

### Archimedes and Numerical Roots

##### Age 14 to 16Challenge 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?

### Deep Roots

##### Age 14 to 16Challenge Level

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

### Rachel's Problem

##### Age 14 to 16Challenge Level

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

### Consecutive Squares

##### Age 14 to 16Challenge Level

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

### Staircase

##### Age 16 to 18Challenge Level

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

### Mod 7

##### Age 16 to 18Challenge Level

Find the remainder when 3^{2001} is divided by 7.

### Rationals Between...

##### Age 14 to 16Challenge Level

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

### Rational Roots

##### Age 16 to 18Challenge Level

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.

### Route to Root

##### Age 16 to 18Challenge Level

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. . . .

### Absurdity Again

##### Age 16 to 18Challenge Level

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