How to Calculate the Volume of the Earth Using Scientific Notation

[BogMonkey]

I got this question on a multiple choice test and I couldn't do it. The question was
"Calculate the volume of the Earth knowing that the mean radius is 6371 km."

What I did was multiply 4*pi*6371^3 then divided the answer by 3 but it didn't yield the correct answer. I got the solutions sheet for the test and here's how the teacher solved the problem

= 4pi * (6.371 * 10^3)^3 / 3
= 4pi * (6.371)^3 * 10^9 / 3

They're the first two steps of his solution but I'm completely lost. How do you magically get a different answer by using scientific notation? I know the laws of indices but I don't understand how I got a different answer because I didn't use scientific notation to represent 6371 km.

 

[Integral]

Since you have not shown us your final answer there is no way we can point you in the right direction. You are correct using scientific notation will not change the result.

 

[BogMonkey]

Sorry here's my answer:
1.08 * 10^15 m^3
heres the correct answer:
1.1 * 10^23 m^3

What I put into my calculator was this 4pi(6371^3) / 3 then to convert to meters I multiplied by 1000. Ah right now I see my mistake. 6371^3 * 1000 is not the same as 6371000^3. I suppose a good rule of thumb to follow is to convert to the appropriate units from the start.

 

[f95toli]

I suppose a good rule of thumb to follow is to convert to the appropriate units from the start.

No, it is not just a good rule of thumb: For "beginner" it is a rule. You should always convert all values to SI before you plug them into an equation.
Once you get more used to the equations you are working with you can start "cheating" a bit, but the "default" should always be to use SI for everything.
That is the best way to avoid making errors.