Help seeing between the lines

Hi guys. I'm currently attending an online course and during the lecture the teacher has solved an equation for some variable 'a'. He showed the following two lines as an explanation but I cannot see between the lines.

51.1 = a * 8000^3
a = 0.998 * 10^-10

My question is how has he arrived at this second line?

I'm guessing this is me not understanding how to convert numbers to the form a * 10 ^ b however I don't know the name of this form nor how to use it.

I understand that the first line can be simplified to:

a = 51.1 / ( 8000^3 )

Which implies:

51.1 / ( 8000 ^ 3 ) == 0.998 * 10^-10

But how?!

Thanks.

Help understanding equation simplication

Hi guys. I'm currently attending an online course and during the lecture the teacher has solved an equation for some variable 'a'. He showed the following two lines as an explanation but I cannot see between the lines.

51.1 = a * 8000^3
a = 0.998 * 10^-10

My question is how has he arrived at this second line?

I'm guessing this is me not understanding how to convert numbers to the form a * 10 ^ b however I don't know the name of this form nor how to use it.

I understand that the first line can be simplified to:

a = 51.1 / ( 8000^3 )

Which implies:

51.1 / ( 8000 ^ 3 ) == 0.998 * 10^-10

But how?!

Thanks.

emptyconvict said:

Hi guys. I'm currently attending an online course and during the lecture the teacher has solved an equation for some variable 'a'. He showed the following two lines as an explanation but I cannot see between the lines.

51.1 = a * 8000^3
a = 0.998 * 10^-10

My question is how has he arrived at this second line?

I'm guessing this is me not understanding how to convert numbers to the form a * 10 ^ b however I don't know the name of this form nor how to use it.

I understand that the first line can be simplified to:

a = 51.1 / ( 8000^3 )

Which implies:

51.1 / ( 8000 ^ 3 ) == 0.998 * 10^-10

But how?!

Thanks.

Click to expand...

By elementary division.

Did you try to put 51.1 / ( 8000 ^ 3 ) in your calculator? What did you get?

Your teacher probably used a calculator!!

In general, the expressing a number in terms of a * 10^b is called scientific notation. Traditionally, it is done in such a way that 1 <= a < 10. You can read more about scientific notation here. That said, you can get the result your instructor got by simply doing the division, as Subhotosh Khan told you. However, rather than simply rely on a calculator, I strongly encourage you to try and understand why the math works, because that will serve you better in the future. Let's work through it a bit, starting with the given:

\(\displaystyle a=\frac{51.1}{8000^3}=51.1\cdot 8000^{-3}\)

Factor 8000 into 20³:

\(\displaystyle a=51.1\cdot \left(20^3\right)^{-3}=51.1\cdot 20^{-9}\)

\(\displaystyle a=51.1\cdot \left(2\cdot 10\right)^{-9}=51.1\cdot 2^{-9}\cdot 10^{-9}=\frac{51.1}{2^9}\cdot 10^{-9}\)

Because the denominator (2^9) is a power of 2, we know it's decimal representation will terminate. So, we can use a calculator to get the exact value.

\(\displaystyle a=0.0998046875\cdot 10^{-9}\)

Remember that for scientific notation, we want the a term to be greater than 1, so let's divide by 100 (or in other words, multiply by 10^-2)

\(\displaystyle a=0.0998046875\cdot 10^{-9}\cdot 10^{-2}=9.98046875\cdot 10^{-11}\)

For whatever reason, your instructor only reduced it to a multiple 10^-10 and left it in improper scientific notation, but hopefully you can see why the principle is the same.