Resources tagged with: Powers & roots

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Broad Topics > Properties of Numbers > Powers & roots

Bina-ring

Age 16 to 18 Challenge Level:

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

The Root of the Problem

Age 14 to 18 Challenge Level:

Find the sum of this series of surds.

Plus or Minus

Age 16 to 18 Challenge Level:

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

Pythagorean Fibs

Age 16 to 18 Challenge Level:

What have Fibonacci numbers got to do with Pythagorean triples?

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 ?

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.

In Between

Age 16 to 18 Challenge Level:

Can you find the solution to this algebraic inequality?

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).

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?

Googol

Age 16 to 18 Short Challenge Level:

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

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?

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.

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 ?

Mod 7

Age 16 to 18 Challenge Level:

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

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?

Cube Roots

Age 16 to 18 Challenge Level:

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

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?

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.

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?

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. . . .

How Many Solutions?

Age 16 to 18 Challenge Level:

Find all the solutions to the this equation.

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?

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.

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?

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?

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?

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!

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?

Deep Roots

Age 14 to 16 Challenge Level:

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

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

Rationals Between...

Age 14 to 16 Challenge Level:

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

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.

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...

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?