How come some units have exponents? ex: cm³?

Question

this might sound dumb but i really don't get why some units of measurement have exponents
need to know why to do my homework...

400 cm³ = ___ L?

400 cm³ = ___ m³ ?

all so confusing to me :/ giving the 10 points for whoever gives best explanation!!! thnanks :/

Answer 1

It turns out to be a useful way of indicating the "type" of thing that is being measured (e.g. length, or area, or volume).

To get an "area" you have to multiply a "length" times a "length". For example, you have to multiply inches times inches. The resulting "inches²" is a way to indicate that you multiplied (inches)(inches) to get the answer.

Likewise for volume:
volume = (cm)(cm)(cm) = cm³

Answer 2

units, like numbers, get multiplied.
Sometimes the product of two units is a third unit.

first: multiplication of units.
You have a prism that measures 3 metre long, 2 metre wide and 1 metre high. What is the volume?
We are used to simply multiply 3 * 2 * 1. In reality, we should multiply the units as well.
3 m * 2 m * 1 m = (3*2*1) * (m*m*m) = 6 m^3 (6 cubic metres).

In mechanics, torque is measured by the force you apply on a lever AND by the length of the lever. Thus, to tighen a bolt, the manufacturer says to use a torque of 100 foot-pound. You can use a 100 foot long spanner and apply a force of one pound, or you can use a 1-foot spanner and apply a force of 100 pounds; as far as the bolt is concerned, the torque it feels is exactly the same.

33.33... pounds * 3 feet = 100 foot-pounds.

Equivalence:

In any system, there are units that are multiple of other units: for example, one gallon = 4 quarts.

In the SI system, the factors are powers of 10.

a cube of 1 cm by 1 cm by 1 cm is 1 cm^3 (also cm³ but many users of "answers" can't do exponents, so I use the caret symbol). It is called a cubic centimeter or a c.c.

1000 cm^3 is called a litre. We can write 1000 cm^3 / L (pronounced one thousand cubic centimetres per litre).

As it happens, one c.c. of pure water has a mass of 1 gram; therefore 1 litre of pure water is 1000 grams (called a kilogram).

400 cm^3 / (1000 cm^3 / L) = (400 / 1000)*(cm^3 * L / cm^3)

The cm^3 cancel out, leaving only L (litres), what we seek

(400/1000) L = 0.4 L

what is a cubic metre?
1 m^3 = 1 m * 1 m * 1 m
we know that one metre = 100 cm, so we rewrite the equation:

1 m^3 = 100 cm * 100 cm * 100 cm = 1,000,000 cm^3

So there are 1 million cubic centimetres in a cubic metre (using pure water, we get 1,000,000 grams per tonne)

400 cm^3 = ?
400 cm^3 / (1,000,000 cm^3 / m^3) = (400/1,000,000)*(cm^3 * m^3 / cm^3) = (400/1,000,000) m^3 = 0.0004 m^3

Products of units:

You heard of F = m a

The force is in Newtons (in the International System), the mass is in kilograms and the acceleration is in metres per second squared -- because every second, the speed increases by a certain number of metres per second; the unit "seconds" is present twice in the denominator, that is why it is squared.

1 N = 1 Kg * 1 m/s^2

Treating units the same way as you would treat numbers, you are allowed to rewrite this:
1 = 1 kg * m / (N * s^2)
or 1 kg = 1 N * s^2 / m

This is sometimes useful in transforming a bunch of units into a much simpler unit.

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So, when units are present many times in the same equation, they get multiplied together, the same as numbers; if the same unit is there three times, you can write it with an exponent of 3.

Answer 3

Well.. cm^3 is an expression of volume.
If you're trying to get the volume of a rectangle, for instance, you're multiplying LxWxH
Because you're taking three measurements and multiplying them together, essentially the unit is multiplied together too:

See:
3mx4mx1m = 3x4x1 but also mxmxm = 12 m^3.

The same goes for area (Squared because length x width, cm(x)cm).

Converting cm^3 to Liters is just a way of converting from the metric system to a different unit of volume (liter).
Converting cm^3 to m^3 is just a matter of decimal placement.
Hope that helped!

Answer 4

That particular exponent shows that the unit is cubed. So, it's not a measure of distance, but of volume. If it were squared instead, it would be a measure of area. (The L isn't cubed because a liter is already a unit of volume.)