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#### Resources tagged with Powers & roots similar to Weekly Challenge 34: Googol:

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Broad Topics > Numbers and the Number System > Powers & roots

### Weekly Challenge 34: Googol

##### Stage: 5 Short Challenge Level:

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

### Archimedes Numerical Roots

##### Stage: 5 Challenge Level:

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

### Em'power'ed

##### Stage: 5 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?

### Equal Temperament

##### Stage: 4 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.

### Smith and Jones

##### Stage: 4 Challenge Level:

Mr Smith and Mr Jones are two maths teachers. By asking questions, the answers to which may be right or wrong, Mr Jones is able to find the number of the house Mr Smith lives in... Or not!

### Ab Surd Ity

##### Stage: 5 Challenge Level:

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

### Fibonacci Fashion

##### Stage: 5 Challenge Level:

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

### The Root of the Problem

##### Stage: 4 and 5 Challenge Level:

Find the sum of this series of surds.

### Mod 7

##### Stage: 5 Challenge Level:

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

### Pythagorean Fibs

##### Stage: 5 Challenge Level:

What have Fibonacci numbers got to do with Pythagorean triples?

### Staircase

##### Stage: 5 Challenge Level:

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

### Negative Power

##### Stage: 4 Challenge Level:

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

### Power Countdown

##### Stage: 4 Challenge Level:

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

### Plus or Minus

##### Stage: 5 Challenge Level:

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

### Giants

##### Stage: 5 Challenge Level:

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

### Fit for Photocopying

##### Stage: 4 Challenge Level:

Explore the relationships between different paper sizes.

### Lost in Space

##### Stage: 4 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?

### Enriching Experience

##### Stage: 4 Challenge Level:

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

### Guesswork

##### Stage: 4 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.

### In Between

##### Stage: 5 Challenge Level:

Can you find the solution to this algebraic inequality?

### Rachel's Problem

##### Stage: 4 Challenge Level:

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

### Rationals Between...

##### Stage: 4 Challenge Level:

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

### How Many Solutions?

##### Stage: 5 Challenge Level:

Find all the solutions to the this equation.

### Route to Root

##### Stage: 5 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. . . .

### Napier's Location Arithmetic

##### Stage: 4 Challenge Level:

Have you seen this way of doing multiplication ?

### Archimedes and Numerical Roots

##### Stage: 4 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?

### Deep Roots

##### Stage: 4 Challenge Level:

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

### Function Pyramids

##### Stage: 5 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?

### Unusual Long Division - Square Roots Before Calculators

##### Stage: 4 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?

### Square Pair Circles

##### Stage: 5 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.

### Perfectly Square

##### Stage: 4 Challenge Level:

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

### Consecutive Squares

##### Stage: 4 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?

### Rational Roots

##### Stage: 5 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.

### Take a Square

##### Stage: 4 Challenge Level:

Cut off three right angled isosceles triangles to produce a pentagon. With two lines, cut the pentagon into three parts which can be rearranged into another square.

### Root to Poly

##### Stage: 4 Challenge Level:

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

### The Root Cause

##### Stage: 5 Challenge Level:

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

### Number Rules - OK

##### Stage: 4 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...