Detailed explanation:

Let's find the diagonal AC of the bottom square, drawn in green below:



Triangle ADC is a right triangle, with right angle ADC so we use the
Pythagorean theorem:

AC² = AD² + DC² 
AC² = 5² + 5²
AC² = 25 + 25
AC² = 50
       __    ____    __  _     _ 
 AC = Ö50 = Ö25×2 = Ö25×Ö2 = 5Ö2
                                                            _
That's the first answer.  So we label that green diagonal 5Ö2



Let's draw the cube's diagonal AB in green also




Triangle ABC is a right triangle with right angle ACB. So we use
the Pythagorean theorem to find AB

AB² = AC² + BC²
         _
AB² = (5Ö2)² + 5²
          _
AB² = 5²(Ö2)² + 25

AB² = 25(2) + 25

AB² = 50 + 25

AB² = 75
       __    ____    __  _     _
 AB = Ö75 = Ö25×3 = Ö25×Ö3 = 5Ö3

Edwin