Alg2.1.26

Find $(56-24\sqrt{5})^{\frac{1}{4}}\,$

Readjusting the index,we get

$((56-24\sqrt{5})^{\frac{1}{2}})^{\frac{1}{2}}\,$

$\sqrt{\sqrt{(56-24\sqrt{5})}}\,$

Rearranging the numbers in the square root

$\sqrt{\sqrt{(56-2(12)\sqrt{5})}}\,$

$\sqrt{\sqrt{(56-2\sqrt{(144)(5)})}}\,$

$\sqrt{\sqrt{(56-2\sqrt{(36)(20)})}}\,$

$\sqrt{\sqrt{(36+20-2\sqrt{(36)(20)}}}\,$

Rearranging the numbers, we get

$\sqrt{\sqrt{((\sqrt{36})^{2}+(\sqrt{20})^{2}-2(\sqrt{36})(\sqrt{20}))}^{2}}\,$

Hence

$\sqrt{\sqrt{36}-\sqrt{20}}\,$

Simplifying further

$\sqrt{6-\sqrt{20}}\,$

Rearranging the numbers in the above,

$\sqrt{6-2\sqrt{(5)(1)}}\,$

Here again doing the rearrangement,we get

$\sqrt{(\sqrt{5}^{2}+(\sqrt{1}^{2})-2\sqrt{(5)(1)}}\,$

$\sqrt{(\sqrt{5}-1)^2}\,$

Hence the result is

$\sqrt{5}-1\,$