Here is another situation for unit conversion

that is sometimes confusing to people. The specific question here is converting 500

inches squared (or square inches) into centimetres squared (or square centimetres depending on

what you call it). The issue here is converting units with exponents

in them. Both inches squared and centimetres squared

are units for measuring surface area, and generally we use units of length with an exponent

of 2 to measure areas just like we use units with an exponent of 3 to measure volumes of

three-dimensional spaces. I’m going to show you a mathematical method

for solving this problem that you can apply to any other problem with exponents on units. So, we are starting with 500 inches squared

and we want to know what the area would be in centimetres squared. The first thing we need is a relationship

between inches and centimetres. Now, you might ask at this point, “Why don’t

we find a relation between inches squared and centimetres squared, if those are the

units we’re actually converting between. And the answer is, you could do this question

that we, if you had a relationship like that available, but I’m going to teach you this

more general method because I think it is more useful, it’s not difficult and you can

use this method with any squared unit or cubed unit or even units with higher exponents,

like say if you had metres to the power of four, if you ever ran into that kind of question.” But anyway, I know the relationship is 1 inch

is equal to 2.54 centimetres. From this relation comes the standard two

conversion factors, 1 inch over 2.54 centimetres and 2.54 centimetres over 1 inch. And I want to use the one with inches on the

bottom so it will cancel. That’s because inches is on the top in the

original amount if you write it as a fraction. So far, this is exactly the same as any unit

conversion problem, but here is where things are going to change. Because my original amount was in inches squared,

I have to square the entire conversion factor before I multiply by it. So I’ll put this whole fraction to the exponent

two, and this is what it looks like. Now, it helps here to remember a little about

how fractions work with exponents. Because the exponent is outside the bracket,

this is, arount the entire fraction, this exponent applies to the entire fraction top

and bottom. So I need to square both the top and the bottom

of this fraction, and the way it works out is the 2.54 squared is 6.45 and centimetres

squared is just centimetres squared, because that’s what happens when you square a unit

or raise a unit to any exponent, really, it’s just… the unit gets put to that exponent. So, what I just did was take this whole top

– 2.54 cm – and it because 2.54 squared and centimetres squared – the 2.54 squared is

6.45 and the centimetres squared is just centimetres squared. And now, doing the same thing with the bottom

of the fraction, I get 1 squared is 1 and inches squared is just inches squared. So, I did bottom the exact same way as the

top, and this is the result. So now, you can see with these two fractions

that inches squared will cancel, and all we have left to do is mulitply the fractions

together. 500 times 6.45 over 1 is equal to 3226, and

the only unit that hasn’t been canceled is centimetres squared, so that’s the answer. So, I’ll just write my final statement stating

that 500 inches is the same as 3226 centimetres squared. And that is the correct answer, but it’s always

a good idea to check and make sure it makes sense. Does it make sense that over 3000 centimetres

squared covers the same area 500 inches squared? And if you don’t have an idea of how big those

areas are, just check over each step, make sure the units cancelled properly (since if

they didn’t cancel, that could be a source of error), and make sure all the multiplying

was done right.