# Multiplying with Scientific Notation

Multiplying numbers that are in scientific notation is fairly simple because multiplying powers of 10 is so 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 10^{5})(2 x 10^{7})

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)(10^{5} x 10^{7})

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 10^{5} by 10^{7}:

10^{5} x 10^{7} = 10^{5 + 7} = 10^{12}

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

8.6 x 10^{12}

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 10^{12}.

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

Because scientific notation uses positive decimals less than 10, when you multiply two of these decimals, the result is always a positive number less than 100. So in Step 4, you never have to move the decimal point more than one place to the left.

This method even works when one or both of the exponents are negative numbers. For example, suppose you want to multiply the following:

(6.02 x 10^{23})(9 x 10^{–28})

1. Multiply 6.02 by 9 to find the decimal part of the answer:

6.02 x 9 = 54.18

2. Multiply 10^{23} by 10^{–28} by adding the exponents:

10^{23} x 10^{–28} = 10^{23 + –28} = 10^{–5}

3. Write the answer as the product of the two numbers:

54.18 x 10^{–5}

4. Because 54.18 is greater than 10, move the decimal point one place to the left and add 1 to the exponent:

5.418 x 10^{–4}

Note: In decimal form, this number equals 0.0005418.

Scientific notation really pays off when you’re multiplying very large and very small numbers. If you’d tried to multiply the numbers in the preceding example the usual way, here’s what you would’ve been up against:

602,000,000,000,000,000,000,000 x 0.0000000000000000000000000009

As you can see, scientific notation makes the job a lot easier.