I need help with calculating the following,$$\sqrt{7+5\sqrt{2}}-\sqrt{3-2\sqrt{2}}$$ i have tried to solve it as $$\sqrt{\left(\sqrt{7+5\sqrt{2}}-\sqrt{3-2\sqrt{2}}\right)^2}$$ but i've come nowhere.
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4I think that the first root should be a cube root rather than a square root - check your source. – Mark Bennet Apr 18 '15 at 16:23
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Another hint:$(1-\sqrt2)^2=3-2\sqrt2$. – Tim Raczkowski Apr 18 '15 at 16:25
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if you have any radical with the form $$\sqrt{(a+b)\pm2\sqrt{a\times b}}$$ you can rewrite this at the form $$\sqrt{a}\pm\sqrt{b}$$
Do the operative form, guess $\sqrt{x\pm2\sqrt{y}}=\sqrt{a}+\sqrt{b}$ then elevate to square and identify terms
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Note that $3-2\sqrt{2}=\sqrt{2}^2-2(\sqrt{2}\times 1)+1^2$ and $7+5\sqrt{2}=(\sqrt{2}+1)^3$ so I concur with Mar Bennet's comment above.
Kim Jong Un
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Wow, its amazing that such an expression with roots within roots and roots of different degrees equates to 1... What type of math concerns itself with finding means to simplify such expressions? – Just_a_fool Apr 18 '15 at 16:31
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@Just_a_fool Keep in mind that this problem was probably built up 'from the other end' - picking two numbers that differ by 1 and then squaring and cubing them. In general if you're wondering about manipulation of complicated radical expressions, the tool you may want to look into more deeply is Galois Theory. – Steven Stadnicki Apr 18 '15 at 16:54