#### Answer

The element's atomic mass number is 56

#### Work Step by Step

Let's assume that we have 1 mole of atoms. We can find the number of atoms along each side of the cube.
$(6.02\times 10^{23})^{1/3} = 8.44\times 10^7~atoms$
We can find the length $L$ of each side of the cube.
$L = (8.44\times 10^7~atoms)(2.27\times 10^{-10}~m)$
$L = 0.01916~m$
We can find the volume of the cube.
$V = L^3$
$V = (0.01916~m)^3$
$V = 7.034\times 10^{-6}~m^3$
We can find the mass of the cube.
$m = \rho~V$
$m = (7950~kg/m^3)(7.034\times 10^{-6}~m^3)$
$m = 0.0559~kg$
We can find the mass $m_a$ of each atom.
$m_a = \frac{m}{6.02\times 10^{23}}$
$m_a = \frac{0.0559~kg}{6.02\times 10^{23}}$
$m_a = 9.236\times 10^{-26}~kg$
We can find the element's atomic mass number.
$A = \frac{m_a}{1.66\times 10^{-27}~kg}$
$A = \frac{9.236\times 10^{-26}~kg}{1.66\times 10^{-27}~kg}$
$A = 56$
The element's atomic mass number is 56.