The solving process is much the same as solving problems with square units, except you have to cube the absolute exponential distance between the two prefixes. I will do the same example problems as in the square unit tutorial, except as cubic problems.

Remember that cubic problems deal with volume.

Here are the problems:

1.00 m^{3}= how many cubic centimeters?

1.00 km^{3}= how many cubic micrometers?

Before going into the solutions, try them out before seeing the answers.

**First problem:** The absolute exponential distance between the base unit and centi- is 10^{2}. Sometimes a teacher may say the exponential distance is 2. Be aware that the 2 is an exponent, NOT that you will be multiplying something by 2. In a cubic situation, we must cube it to 10^{6}. The reason we do this is because there are three centimeter measurements, one for each of the three sides of the cube.

Let me anticipate a problem: you read the above and say "Wait, the cube of 2 is 8, isn't it?" You are right because (2)^{3} = 2 x 2 x 2 = 8, but we are dealing with an exponential value of 2, not just the number 2. What gets cubed is 10^{2}, not 2. We would write (10^{2})^{3} = 10^{2 x 3} = 10^{6}.

Now, to finish the problem:

1.00 m^{3}= 1.00 x 10^{6}cm^{3}

**First problem:** Between kilo- and micro- is an absolute distance of 10^{9}. Our technique calls for the exponential value to be cubed, so (10^{9})^{3} = 10^{27}. The direction of change is going from the larger prefix to the smaller prefix, so the sign of the exponent in the answer will be positive. Our answer is:

1.00 km^{3}= 1.00 x 10^{27}μm^{3}

Video: How to Convert between Cubic Meters and Liters.

How to Convert from kg/m^{3} to g/L

My examples above were going from the larger prefix to the smaller. That meant there were lots of the smaller prefix in one big prefix. The first problem in each set goes the opposite direction. Ask yourself: how many of a larger prefix are there in one small prefix? Answer = less than one, so the sign on the exponent will be negative.

Having said that, watch out for a problem where there is an exponent already in the problem and the negative exponent change fails to overwhelm the exponent from the problem. In that case, there will be a positive exponent in the answer.

**Cube unit problems**

1) 4310 cubic centimeters. Convert to m^{3}.

2) 86.3 cubic centimeters. What is volume in mm^{3}?

3) 5.94 x 10^{10} mm^{3}. Convert to dm^{3}.

Here's another example: A metal cube is 2.50 cm on a side. Calculate the volume in m^{3}.

**Solution:**

1) Convert 2.50 cm to meters:

(2.50 cm) (1 m / 100 cm) = 0.0250 m

2) Multiply to obtain the volume of the cube:

(0.0250 m) (0.0250 m) (0.0250 m) = 1.56 x 10¯^{5}m^{3}(to three sig figs)

3) Here's an alternate solution path:

(2.50 cm) (2.50 cm) (2.50 cm) = 15.625 cm^{3}(15.625 cm

^{3}) (1 m / 100 cm)^{3}= 15.625 x 10¯^{6}m^{3}15.625 x 10¯

^{6}m^{3}= 1.56 x 10¯^{5}m^{3}(to three sig figs)