Sometimes on the news you will hear about the federal deficit being close to 34 trillion dollars. But what does that number look like? To write out 34 trillion would look like 34,000,000,000,000. That’s a lot of zeros. It’s also a very large and inconvenient number to write. Unless we use scientific notation.

Scientific notation is a way of using exponents and powers of 10 to write very large or very small numbers. In general, a number written in scientific notation looks like

• The base number, a, must be a number one or larger but less than 10 ( ).
• The exponent, b, can be either positive or negative.

The most common problems we see with scientific notation involve changing from standard notation to scientific notation and vice versa. Let’s look at these types of problems before moving on to operations with scientific notation.

We’ll start by writing the deficit amount in scientific notation.

Let's Practice:

1. Write 34,000,000,000,000 in scientific notation.

For a number to be written in scientific notation, we need a number between 1 and 10 and a power of 10. To determine the number between 1 and 10, look at the original number and determine where you can place a decimal point to create a number between 1 and 10. For this problem, we can place the decimal point between 3 and 4 to get 3.4 which is between 1 and 10.

But what about all those zeros?

If the decimal point is between the 3 and 4, how many places would it need to be moved to get to the end of the number? If we move the decimal 13 places, we end up where we started. The 13 then becomes our exponent. So 34,000,000,000,000 written in scientific notation would be

Notice that the exponent is positive because in determining our exponent we moved the decimal 13 places to the right to get back to our original number.

2. Write 0.000000000294 in scientific notation.

Once again we want to begin by creating a number between 1 and 10 which would be 2.94. We would need to move the decimal point 10 places to the left to get it back to where it originally started. So 0.000000000294 written in scientific notation is

In this case, we used a negative exponent because we would need to move the decimal point 10 places to the left to get back to the original number.

3. Write 2.7 x 10¹⁵ in expanded notation.

In this case, we start with 2.7 and move the decimal point 15 places to the right (because the exponent is positive). You may be wondering how we can move the decimal point 15 spaces when after one space we are at the end of our number. To keep moving the decimal point, keep adding zeros.

2,700,000,000,000,000

Once we know how to go back and forth between standard notation and scientific notation, it is very common to perform operations on numbers that are written in scientific notation. Doing this involves making use of the rules of exponents.

4. 

To find the answer to this problem, we can multiply 9.3 and 6.2 and get 57.66.

We can also multiply to get .

So putting these two pieces together we get

But remember that to be in scientific notation, the number must be between 1 and 10. Clearly 57.66 is not between 1 and 10. But if we move the decimal so that the number becomes 5.766, we would be OK. But we can’t move the decimal point without affecting the exponent attached to the 10. Since we need to move the decimal point one space the answer to the problem in scientific notation is