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

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Explore the relationships between different paper sizes.

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

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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.

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

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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.

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

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

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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

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

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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?

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

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

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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?

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

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

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

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

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However did we manage before calculators? Is there an efficient way to do a square root if you have to do the work yourself?

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

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What have Fibonacci numbers got to do with Pythagorean triples?

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

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

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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?

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

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

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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.

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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?

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

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

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

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

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

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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.

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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. . . .

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