Multiplying and dividing by powers of ten, writing very large and very small numbers in scientific notation, converting back to standard form, and multiplying and dividing in scientific notation.
Moving the decimal point
Multiplying by \(10\) moves the decimal point one place to the right; dividing by \(10\) moves it one place to the left. Powers of ten, including negative ones, describe any number of moves:
A light-year is about \(9{,}460{,}000{,}000{,}000\) kilometers. A dust mite is about \(0.0003\) meters long. Numbers like these are hard to read and easy to miscopy, so scientists write them differently.
So \(9{,}460{,}000{,}000{,}000 = 9.46 \times 10^{12}\) and \(0.0003 = 3 \times 10^{-4}\).
Going back to standard form just reverses the process: \(4.2 \times 10^5 = 420{,}000\) and \(1.9 \times 10^{-3} = 0.0019\).
Multiplying and dividing
Group the decimal parts together and the powers of ten together, then use the laws of exponents.
To add or subtract, the powers of ten must match first: \(5 \times 10^6 + 3 \times 10^5 = 50 \times 10^5 + 3 \times 10^5 = 53 \times 10^5 = 5.3 \times 10^6\).
Practice
Quick check
Which number is written in scientific notation?
The first factor must be at least \(1\) and less than \(10\). Only \(1.07\) fits.
Quick check
What is \(0.00052\) in scientific notation?
Move the decimal point \(4\) places right to get \(5.2\). The number is less than \(1\), so the exponent is \(-4\).
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