When you need to multiply very large or very small numbers, scientific notation makes the job much simpler. The process breaks down into two distinct parts: handling the decimal numbers and handling the powers of ten. It's a divide-and-conquer approach to arithmetic.

First, you multiply the coefficients—the decimal numbers at the front. Then, you handle the powers of ten. Because the bases are the same (they're all 10), you can simply add their exponents together. This is a direct application of the product rule for exponents.

Let's see how this works with an example. Suppose we want to multiply $3.1 \times 10^5$ by $2.0 \times 10^3$.

This visual breakdown shows how the two parts of the numbers are handled separately before being recombined for the final product. This method is especially powerful in fields like chemistry, where you might calculate the mass of a substance containing a huge number of molecules, often involving Avogadro's number .

The rule doesn't change when negative exponents enter the picture. You still multiply the coefficients and add the exponents. The only difference is that you'll be doing algebraic addition, which might involve negative numbers.

For instance, let's multiply $4.5 \times 10^8$ by $2.0 \times 10^{-3}$.

What if both exponents are negative? The process remains the same. Consider multiplying $5.0 \times 10^{-4}$ by $3.0 \times 10^{-5}$.

Note that in this last example, the final coefficient is 15.0, which is not between 1 and 10. For now, we are just focusing on the raw product. In the next section, we'll cover how to adjust, or normalize, such an answer to put it back into proper scientific notation form.

Before moving on, let's test your understanding of multiplying coefficients and adding exponents.

Mastering this two-step process is the key to efficiently handling calculations with scientific notation.