How to Multiply & Divide in Scientific Notation

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  • 0:03 Scientific Notation in…
  • 1:07 Review of Scientific Notation
  • 2:34 Multiplying and Dividing
  • 3:59 Example
  • 5:47 Negative Digits and Exponents
  • 6:51 Lesson Summary
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Lesson Transcript
Instructor: Thomas Higginbotham

Tom has taught math / science at secondary & post-secondary, and a K-12 school administrator. He has a B.S. in Biology and a PhD in Curriculum & Instruction.

Scientists, engineers, and other people who work with really big or really small numbers often choose to express those numbers in scientific notation. In this lesson, we'll learn how to multiply and divide in scientific notation.

Scientific Notation in Real Life

Imagine you have a special kind of flashlight that causes the light to travel around the circumference of the earth, instead of heading straight off into space. If you turned it on now, by the time you finish reading this sentence, it would have traveled around the earth somewhere around fifty times, or over seven times per second! Now that's fast - approximately 186,000 miles per second.

The speed of light is a great way to explore how to manipulate numbers using scientific notation, not only because of the enormity of distance that light can travel, but because it's just pretty darned interesting. For example, you're more than a little psyched at how fast your minivan is, so much so that you've named it 'Flash.' You want to compare its top speed to the speed of light. You're pretty sure they're close, but need to make sure.

When calculating the speed of your minivan, you'll find that your numbers may get very large very quickly, which will be a perfect time to multiply and divide in scientific notation. In performing these calculations, you'll be trying to answer the question: How many times faster than my minivan is the speed of light?

Review of Scientific Notation

Scientific notation is a way to represent really large or really small numbers in a way that's a little more digestible for most people. It takes advantage of the ease of multiplying and dividing by 10s, like 100s, 1000s etc. For example, 186,000 in scientific notation is 1.86 x 10^5, while 0.00000000006501 in scientific notation is 6.501 x 10^ -11. The first term in each example, or 1.86 and 6.501 respectively, is called the digit term. The second term in each example is called the exponent term.

Smaller, everyday numbers can be represented in scientific notation as well, such as 178 and 3.14, which are 1.78 x 10^2 and 3.14 x 10^0.

Besides knowing how to represent numbers in scientific notation, there's one other skill we need to review before comparing Flash to the speed of light: how to compare two quantities, or in this case, speed. Let's assume that a turtle at top speed can move at 5 miles per hour, while the hare can move at 45 miles per hour. To find out how many times faster the hare is, simply divide 45 mph by 5 mph. The result is 9, so the hare is nine times faster than the turtle.

Multiplying and Dividing in Scientific Notation

To multiply in scientific notation, you'll need to follow these three steps:

  1. Multiply the digit terms
  2. Add the exponents of the exponent terms
  3. Put the results found in the first two steps together
Let's try a multiplication example:

(3 x 10^15) * (1.5 x 10^12)

  1. First we multiply our digit term, so: 3 * 1.5 = 4.5
  2. Next, we add the exponent terms together, meaning: 15 + 12 = 27
  3. Finally, we combine our results to get our answer, which is: 4.5 x 10^27 or 4,500,000,000,000,000,000,000,000,000

To divide in scientific notation, you'll follow a similar set of rules:

  1. Divide the digit terms
  2. Subtract the exponents of the exponent terms
  3. Put the results found in the first two steps together
Let's try a division example

(3 x 10^15) / (1.5 x 10^12)

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