# algebra proof

• August 22nd 2009, 03:11 PM
mms
algebra proof
prove that $
2a\sqrt {\frac{1}
{{6 + 4\sqrt 3 }}} = a\left( {3^{1/4} - 3^{ - 1/4} } \right)

$
• August 22nd 2009, 05:48 PM
Soroban
Hello, mms!

This requires Olympic-level gymnastics . . .

Quote:

Prove that: . $2a\sqrt {\frac{1}
{6 + 4\sqrt 3 }} \:=\: a\left(3^{\frac{1}{4}} - 3^{ -\frac{1}{4}}\right)
$

The left side is: . $\frac{2a}{\sqrt{6 + 4\sqrt{3}}}$ .[1]

Note that: . $6 + 4\sqrt{3} \:=\:\sqrt{3}(2\sqrt{3}+4) \;=\;\sqrt{3}(\sqrt{3}+1)^2$

. . . . . . Hence: . $\sqrt{6+4\sqrt{3}} \;=\;\sqrt[4]{3}(\sqrt{3}+1)$

Then [1] becomes: . $\frac{2a}{\sqrt[4]{3}(\sqrt{3}+1)}$

Multiply by $\frac{\sqrt{3}-1}{\sqrt{3}-1}\!:\quad \frac{2a}{\sqrt[4]{3}(\sqrt{3}+1)}\cdot\frac{\sqrt{3}-1}{\sqrt{3}-1} \;=\;\frac{2a(\sqrt{3}-1)}{\sqrt[4]{3}\,(2)} \;=\;\frac{a(\sqrt{3}-1)}{\sqrt[4]{3}}$

Therefore: . $\frac{a(3^{\frac{1}{2}} - 1)}{3^{\frac{1}{4}}} \;=\;a\left(3^{\frac{1}{4}} - 3^{-\frac{1}{4}}\right)$

• August 22nd 2009, 09:51 PM
QM deFuturo
Quote:

Originally Posted by Soroban
Hello, mms!

This requires Olympic-level gymnastics . . .

Pretty impressive. How did you determine this:
$(2\sqrt{3}+4) \;=\;(\sqrt{3}+1)^2$

It's true, but I would not have been able to figure that out, or even realize I would have to do that for this proof!
• August 23rd 2009, 12:44 AM
Hello QM deFuturo
Quote:

Originally Posted by QM deFuturo
Pretty impressive. How did you determine this:
$(2\sqrt{3}+4) \;=\;(\sqrt{3}+1)^2$

It's true, but I would not have been able to figure that out, or even realize I would have to do that for this proof!

Note that:

• $(a+b)^2 = a^2 + 2ab + b^2$

and

• $(\sqrt3)^2 = 3$

Then notice that $2\sqrt3+4 = 3 + 2\sqrt3 + 1 = (\sqrt3)^2 + 2\sqrt3 + 1$.

Put $a = \sqrt3$ and $b = 1$ in the expansion of $(a+b)^2$, and you're there. I'm afraid this sort of cleverness only comes with years of practice. I (and I suspect Soroban also) have been doing these things since Noah was a boy.