Resources tagged with: Powers & roots

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Broad Topics > Properties of Numbers > Powers & roots

Rational Roots

Age 16 to 18
Challenge 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.

Golden Eggs

Age 16 to 18
Challenge Level

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

Absurdity Again

Age 16 to 18
Challenge Level

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

Plus or Minus

Age 16 to 18
Challenge Level

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

In Between

Age 16 to 18
Challenge Level

Can you find the solution to this algebraic inequality?

Square Pair Circles

Age 16 to 18
Challenge 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.

Cube Roots

Age 16 to 18
Challenge Level

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

How Many Solutions?

Age 16 to 18
Challenge Level

Find all the solutions to the this equation.

Fibonacci Fashion

Age 16 to 18
Challenge Level

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

Irrational Arithmagons

Age 16 to 18
Challenge Level

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

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?

Giants

Age 16 to 18
Challenge Level

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

Pythagorean Fibs

Age 16 to 18
Challenge Level

What have Fibonacci numbers got to do with Pythagorean triples?

The Root of the Problem

Age 14 to 18
Challenge Level

Find the sum of this series of surds.

Em'power'ed

Age 16 to 18
Challenge 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 18
Challenge Level

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

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

Mod 7

Age 16 to 18
Challenge Level

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

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.

Enriching Experience

Age 14 to 16
Challenge Level

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

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?

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?

Googol

Age 16 to 18 Short
Challenge Level

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

Rationals Between...

Age 14 to 16
Challenge Level

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

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?

Number Rules - OK

Age 14 to 16
Challenge Level

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

Staircase

Age 16 to 18
Challenge Level

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

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?

Fit for Photocopying

Age 14 to 16
Challenge Level

Explore the relationships between different paper sizes.

Bina-ring

Age 16 to 18
Challenge Level

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

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 $1+\sqrt 2+\sqrt 3$.

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 ?

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?

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!

Double Trouble

Age 14 to 16
Challenge Level

Simple additions can lead to intriguing results...

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.

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.

Archimedes Numerical Roots

Age 16 to 18
Challenge Level

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 16
Challenge Level

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

Route to Root

Age 16 to 18
Challenge 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. . . .

Function Pyramids

Age 16 to 18
Challenge 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?