Posted in ask uncle colin, surds.

Dear Uncle Colin How does $\sqrt{9 - \sqrt{17}} = \frac{\sqrt{34}-\sqrt{2}}{2}$? I tried applying a formula, but I couldn't make it work. - Roots Are Dangerous, It's Chaotic A-Level Simplification Hi, RADICALS, and thank you for your message! Square roots of square roots are not usually trivial, but this one can

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Posted in surds.

The estimable @solvemymaths tweeted, some time back: hmm, perhaps I'll keep this one as "sin(22)" pic.twitter.com/cT5IHonoyb — solve my maths (@solvemymaths) January 16, 2016 A sensible option? Perhaps. But Wolfram Alpha is being a bit odd here: that's something that can be simplified significantly. (One aside: I'm not convinced that

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Posted in number theory, surds.

Funny thing. Type $(2 + \sqrt{3})^{20}$ into Wolfram Alpha. (Or, if you're really lazy, click this link.). It's 274,758,382,273.999999999996 or so. The higher the power you pick, the closer $(2 + \sqrt{3})^n$ gets to an integer value -- although it never quite gets there, because $\sqrt{3}$ is irrational. So, how

Read More →A student asks: How could I simplify a sum like $(\sqrt 3+\sqrt 2)(\sqrt 3-\sqrt 2)$? Great question! The trick is to treat it like it's an algebraic bracket, like this: $(x + y)(x - y) = x^2 + yx - xy - y^2$ But then you've got $+yx -xy$ in

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