Why are there exponents for Area, Volume and other quantities?

Question

Why are there exponents for Area, Volume and other quantities?

In a problem where there is a rectangle that is 3 meters in length and 4 in width the area would be 12²m. I've always thought of "squared" as defining a number as the area of a shape, object or location. 12²m looks like a unfinnished equation to me. If I multiply 12 with 12 to remove the exponent I receive 144m which isn't close to whats inside a 3m x 4m rectangle.

For another example, this equation where we're solving for "h:" 36.8*20*h=1472³
Multiplying 36.8m with 20m would give us 736².
Now we're down to 736² * h = 1472³.

To solve the last part, since it has exponents, I would firstly attempt canceling out the exponents by subtraction, addition, multiplication or division depending on the problem and is the exponent has exponents. In my first example exponents were just for labeling it was an area and in my second it signified two things were multiplied (36.8*20 = 732"²"). For solving it, how could you cancel out an exponent if the numbers multiplied to achieve one of them differ in value (rather than just the number of times numbers were multiplied "²", "³") from what was used to achieve the other(s)? Wouldn't that mean I couldn't cancel these out without prior information? If they were just used to show the numbers are area and volume then how would you cancel out quantities? Lastly, if they just stand for how many times x is present as in x²,x³, etc, why are they used in correlation with shapes quantities?

Gratitude goes to whoever can assist me in overcoming this confusion.

Thanks

Answer 1

Hi Daniel,

     I never really understood this when I was young either. The thing that helped me was when I truly understood what area and volume meant.

 

A meter, inch, foot mile,.....are all linear measurements. They measure the length of a line segment. This is good for perimeter, or I want to put a fence alongside my driveway, or I want to hang a clothesline between two poles....examples are endless but they all involve a one dimensional distance, from pt A to Pt B.

 

Area is covering up a surface in a 2 dimensional figure...carpeting a room, painting a wall, buying sod to put down a new lawn....  In order to cover up a space, you couldn't use line segments so you can't use linear measurements like feet or centimeters. In order to cover up a figure, you must put down squares that are next to each other. The dimensions of those squares are 1 in by 1 in or 1 mile by 1 mile......which makes the unit a square with a side that is 1 mile long which we abbreviate to 1 sq, mile....or 1 mile^2.

 

Volume consists of filling up 3 dimensional figures with a unit. Now a line segment won't do and not even a flat square is going to fill up anything. A flat square CAN cover up the sides of the figure but it can't fill it up. So square units can be used for Surface area but not for volume. For volume, we need little cubes having dimensions 1 by 1 by 1.....like a die.....you know, two dice but one die. You can take these figures and stack them so that you can fill up the figure. So you need cubes, making the units cubic inches, cubic yds,,,,,ft^3 to fill something up for volume.

 

A weird question for you. How many square inches are there in a square foot?.......Most people say 12 and they're wrong.

 

A square foot is a square having dimension 1 ft by 1 ft. If I convert BOTH dimensions into inches I get 

12 in by 12 in. If you drew lines through the middle of your square foot, you would then have 144 squares which are 1 in by 1 in or 144 sq in. Do you understand that square inches are actually squares? Cubic inches ARE ACTUALLY PHYSICALLY cubes.

 

I never really understood this and nobody every explained this accurately enough for me to understand. I hope you can follow what I am trying to say.

Answer 2

Hi Daniel,

 

This is an excellent question which I will attempt to answer.

 

At the heart of the matter, you are confusing exponents with units of measurement.  For example, 12^3 is a shorthand representation of 12*12*12 = 1728.  But when you see m^2, that represents a square meter, which is a unit of measurement.  Similarly, m^3 is a cubic meter.  M^2 and m^3 are just the way that mathematicians have chosen to represent these units of measurement.  The reason is because a square is two-dimensional and a cube is three-dimensional, so if you try to visualize those numbers it will hopefully make sense.  

 

You can't manipulate units of measurement in the same way that you can exponents attached to numbers.  I could easily do (12^2 / 12^3) = 1 / 12, but if I did (m^2 / m^3), the answer would be incomprehensible.  What do you get when you divide a square by a cube?