Question 1015275
I assume your cube's diagonal is not a space diagonal
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let e1 be an edge of cube 1, then volume of cube 1 is e1^3
We know that e1 is the diagonal of cube 2, let e2 be an edge of cube 
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now use the Pythagorean Theorem
e1^2 = e2^2 + e2^2
2e2^2 = e1^2
e2 = e1/sqrt(2)    
volume of cube 2 is e1^3 / (sqrt(2))^3 = e1^3 / (2sqrt(2))
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the ratio(R) of their volumes is e1^3 / (e1^3 / (2sqrt(2)))
R = (e1^3 * 2sqrt(2)) / e1^3 = 2sqrt(2)
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The other tutor's solution uses the space diagonal, I did not use that since the problem stated "diagonal" only.
It should be clear that a cube has two types of diagonals, one is on a face and the other is inside - called a space diagonal.