### Video Transcript

Find the surface area of one face
of a cube given that its volume is 27 over 125 cubic centimeters.

We begin by recalling that a cube
of side length 𝑙 centimeters will have a volume of 𝑙 cubed cubic centimeters. This means that in this question,
𝑙 cubed is equal to 27 over 125. We note that both 27 and 125 are
perfect cubes, since three cubed is equal to 27 and five cubed is equal to 125. We can therefore rewrite our
equation as 𝑙 cubed is equal to three cubed over five cubed.

In order to solve this, we take the
cube root of both sides. Recalling that if 𝑎 and 𝑏 are
integers and 𝑏 is nonzero, then the cube root of 𝑎 cubed over 𝑏 cubed is equal to
𝑎 over 𝑏. So, the right-hand side of our
equation simplifies to three over five, or three-fifths. Therefore, we conclude that the
side length of a cube with a volume of 27 over 125 cubic centimeters is three over
five centimeters. It is also worth noting that we
could write this in decimal form of 0.6 centimeters.

Now that we have the side length of
the cube, we can determine the surface area of one face. Since each face of a cube is a
square, we will square the side length to find its surface area, that is, 𝑙 squared
square centimeters. This means that the surface area of
one face is three over five squared, which is equal to three over five multiplied by
three over five.

So, the surface area of one face of
a cube given its volume is 27 over 125 cubic centimeters is nine over 25 square
centimeters.