# How do you simplify (sqrt(7) + sqrt(5))^4 + (sqrt(7) - sqrt(5))^4?

Feb 7, 2015

Okay i should thank you for such a question

OK lets start

Call $\sqrt{7} = \sqrt{a} \mathmr{and} \sqrt{5} = \sqrt{b}$

So lets rewrite you equation as ${\left\{{\left(\sqrt{a} + \sqrt{b}\right)}^{2}\right\}}^{2} + {\left\{{\left(\sqrt{a} - \sqrt{b}\right)}^{2}\right\}}^{2}$

So lets take the first part and simplify

${\left(\sqrt{a} + \sqrt{b}\right)}^{2} = a + 2 \sqrt{a b} + b$

So lets substitute $7 + 5 + 2 \sqrt{35} = 12 + \sqrt{35}$

Square this gain and you will get that${\left\{{\left(\sqrt{a} + \sqrt{b}\right)}^{2}\right\}}^{2} = 144 + 35 + 24 \sqrt{35} = 179 + 24 \sqrt{35}$

Similarly repeat the above steps for ${\left\{{\left(\sqrt{a} - \sqrt{b}\right)}^{2}\right\}}^{2}$

After substituting you should get that the above equation is $= 179 - 24 \sqrt{35}$

So finally lets put the puzzle together

$h e n c e {\left\{{\left(\sqrt{a} + \sqrt{b}\right)}^{2}\right\}}^{2} + {\left\{{\left(\sqrt{a} - \sqrt{b}\right)}^{2}\right\}}^{2} = 179 + 24 \sqrt{35} + 179 - 24 \sqrt{35}$

 (sqrt(7) + sqrt(5))^4 + (sqrt(7) - sqrt(5))^4 = 358

Hope that this is what you wanted if this is what you wanted please write down in the comments