Now that we’ve mastered converting between standard and scientific notation, let’s explore how this skill can be applied in real-world scenarios.
Let’s dive into some practical exercises to see scientific notation in action, helping us solve problems efficiently in science, engineering, and beyond.Suppose we are asked to calculate the number of atoms in 1 L of water. Each water molecule contains 3 atoms (2 hydrogen and 1 oxygen).The average drop of water contains around [latex]1.32\times {10}^{21}[/latex] molecules of water and 1 L of water holds about [latex]1.22\times {10}^{4}[/latex] average drops.Therefore, there are approximately [latex]\left(3\right)\cdot\left(1.32\times {10}^{21}\right)\cdot \left(1.22\times {10}^{4}\right)=\left(3\cdot1.32\cdot1.22\right)\times\left({10}^{4}\cdot{10}^{25}\right)\approx 4.83\times {10}^{25}[/latex] atoms in 1 L of water.
We simply multiply the decimal terms and add the exponents. Imagine having to perform the calculation without using scientific notation!
Note: How are we are able to simply multiply the decimal terms and add the exponents? What properties of numbers enable this?
Recall that multiplication is both commutative and associative. That means, as long as multiplication is the only operation being performed, we can move the factors around to suit our needs. Lastly, the product rule for exponents allows us to add the exponents on the base of [latex]10[/latex].
Perform the operations and write the answer in scientific notation.