How to Multiply with Scientific Notation

Multiplying numbers that are in scientific notation is fairly simple because multiplying powers of ten is easy. Here's how to multiply two numbers that are in scientific notation:

1. Multiply the two decimal parts of the numbers.

   Suppose you want to multiply the following: (4.3 x 10⁵)(2 x 10⁷)

   Multiplication is commutative, so you can change the order of the numbers without changing the result. And because of the associative property, you can also change how you group the numbers. Therefore, you can rewrite this problem as (4.3 x 2)(10⁵ x 10⁷).

   Multiply what's in the first set of parentheses — 4.3 x 2 — to find the decimal part of the solution: 4.3 x 2 = 8.6.

2. Multiply the two exponential parts by adding their exponents.

   Now multiply 10⁵ x 10⁷: 10⁵ x 10⁷ = 10^{5 + 7} = 10¹²

3. Write the answer as the product of the numbers you found in Steps 1 and 2.

   8.6 x 10¹²

4. If the decimal part of the solution is 10 or greater, move the decimal point one place to the left and add 1 to the exponent.

   Because 8.6 is less than 10, you don't have to move the decimal point again, so the answer is 8.6 x 10¹².

   Note: This number equals 8,600,000,000,000.

This method works even when one or both of the exponents are negative numbers. For example, if you follow the preceding series of steps, you find that (6.02 x 10²³)(9 x 10^–28) = 5.418 x 10^–4. Note: In decimal form, this number equals 0.0005418.