WEBVTT
00:00:01.770 --> 00:00:11.870
Find the surface area of one face of a cube given that its volume is 27 over 125 cubic centimeters.

00:00:13.090 --> 00:00:21.700
We begin by recalling that a cube of side length 𝑙 centimeters will have a volume of 𝑙 cubed cubic centimeters.

00:00:22.600 --> 00:00:29.600
This means that in this question, 𝑙 cubed is equal to 27 over 125.

00:00:30.410 --> 00:00:42.270
We note that both 27 and 125 are perfect cubes, since three cubed is equal to 27 and five cubed is equal to 125.

00:00:43.190 --> 00:00:50.000
We can therefore rewrite our equation as 𝑙 cubed is equal to three cubed over five cubed.

00:00:50.990 --> 00:00:55.010
In order to solve this, we take the cube root of both sides.

00:00:55.880 --> 00:01:05.710
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.

00:01:06.560 --> 00:01:12.520
So, the right-hand side of our equation simplifies to three over five, or three-fifths.

00:01:13.470 --> 00:01:23.810
Therefore, we conclude that the side length of a cube with a volume of 27 over 125 cubic centimeters is three over five centimeters.

00:01:24.930 --> 00:01:31.440
It is also worth noting that we could write this in decimal form of 0.6 centimeters.

00:01:32.610 --> 00:01:38.530
Now that we have the side length of the cube, we can determine the surface area of one face.

00:01:39.450 --> 00:01:49.970
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.

00:01:51.120 --> 00:02:01.830
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.

00:02:02.540 --> 00:02:15.060
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.