Question

WHat is the simplification of 5.4 x 10^12/ 1.2 x 10^3 written in scientific notation?

Answer 1

Divide \(5.4/ 1.2\), what do you get? That will be the provisional first part of the answer.

Next, take the exponents and subtract to get the exponent for the provisional second part of the answer. Here, we have \(10^{12} / 10^3\) so the exponent will be \(10^{12-3} = 10^9\).

Now I said "provisional" for both of those parts. Here's why: if the first part comes out to be a number \(x\) such that \(1 \le |x| \lt 10\), we're all done. However, if it falls outside of that range, we need to "normalize" it by repeatedly multiplying or dividing by 10 until it does fall into that range.

An example:

\[\frac{2.0 * 10^4}{5.0*10^2} = 0.4 * 10^2\]by the procedure given above. However, we don't want to have \(0.4\) leading off a number in scientific notation. How do we get it into the desired range? Well, we multiply it by 10. Now if we multiply the first part by 10, we need to divide the second part by 10 so that our value remains the same. We do that by subtracting one from the exponent. That makes our answer be \[\frac{2.0 * 10^4}{5.0*10^2} = 0.4 * 10^2 = 0.4*10 * 10^{2-1} = 4.0 * 10^1\]

If on the other hand we had done this problem:
\[2.0*10^4*5.1*10^2 = 10.2*10^6\]we would need to divide our first part by 10 and correspondingly increase the exponent by 1
\[2.0*10^4*5.1*10^2 = 10.2*10^6 = 10.2/10 * 10^{6-1} = 1.02*10^5\]

In the problem you've been assigned, the first part comes out to be between 1 and 10, so you don't need to normalize it, but it is important to understand how to do so. Many problems are not so accommodating :-)