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3.14626436994

sqrt of 24 = 4.89897948557 

add 5 = 9.89897948557

The find the sqrt of 9.89897948557 = 3.14626436994

I am not sure if this is right...

hmm i think guest wanted it to be simplified exactly (like in simpler terms of radicals)...

here's my try at this problem:

$\sqrt{5+\sqrt{24}}=\sqrt{5+2\sqrt{6}}$

so we can express $\sqrt{5+2\sqrt{6}}$  in the form of $(a+b)^2$

so we know that $a^2+b^2=5 \text{ and } 2ab=\sqrt{6}$, so it's really easy to figure out that $a=\sqrt{2} \text{ and } b=\sqrt{3}$

so this would become $\sqrt{(\sqrt{2}+\sqrt{3})^2}$

which means the equation is really just equal to $\boxed{\sqrt{2}+\sqrt{3}}$

i hope this helped you! :)