The diagonals of a square are 7√2 inches long. What is the length of each side of the square?

A square is a special type of rhombus.  And the diagonals of all rhombuses bisect one another at right angles.

So the diagonals divide the rhombus into 4 congruent right angled triangles.

the short sides are  each  $${\frac{{\mathtt{7}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{2}}}}}{{\mathtt{2}}}}$$   inches long.  And the side of the rhombus is the hypotenuse

$$\\h^2=\left(\frac{7\sqrt2}{2}\right)^2+\left(\frac{7\sqrt2}{2}\right)^2\\\\ h^2=2\times \left(\frac{7\sqrt2}{2}\right)^2\\\\ h^2=2\times \frac{49\times 2}{4}\\\\ h^2=49\\\\ h=7\;inches$$

The diagonal of a square is √2 times as long as the length of a side.

So, dividing the diagonal length by √2, we have that the side is just 7 inches.