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A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

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Ab Surd Ity

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

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Absurdity Again

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

The Root of the Problem

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem

This problem offers students an opportunity to practise manipulating surds in the denominator, and highlights the importance of not rounding off prematurely, as by keeping surds in the calculation and simplifying as much as possible, a pleasing answer emerges that might be hidden if students used a calculator and rounded their answers along the way.

Possible approach

Invite students to use spreadsheets to sum parts of the sequence: $$\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + ... +\frac{1}{ \sqrt {99}+ \sqrt{100}}.$$

We hope students will be surprised when they notice that at various points in the sequence, the sum is a whole number, and that they will conjecture about when this happens and wish to explain it. They may need reminding about techniques to rationalise the denominator.

Key questions

For which values of $n$ does the series give whole numbers?
Why might that be?
Can we express $\frac{1}{\sqrt{n}+\sqrt{n+1}}$ in a way that the surds are in the numerator rather than the denominator?

Possible extension

Irrational Arithmagons and Ab Surd Ity are both challenging problems involving the manipulation of surds.

Possible support

Students could start by finding an expression for $\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}}$ and then add subsequent terms.