# Scientific Notation: Practice Problems

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• 0:01 Scientific Notation
• 2:25 Problem 1
• 3:50 Problem 2
• 5:50 Lesson Summary

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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

Scientific notation has a lot of exponents, but it's really not that bad - and it's really convenient for working with awkwardly big or small numbers! Get some practice here.

## Scientific Notation

Do you know what the mass of a proton is?

0.0000000000000000000000000016726 kilograms

Do you know how many scientists want to spend their time writing all those 0s out every time?

That's right: none of them.

That's why we have scientific notation. Scientific notation is a system for abbreviating very large or very small numbers. Instead of that whole mess of zeroes, you could just write:

1.6726 x 10^ -27 kg

This makes it much less likely that you'll make a mistake with the number of zeroes and accidentally mess up your calculations - and it's a lot easier to read.

Here's how scientific notation works:

All numbers in scientific notation are expressed in the form:

a x 10^b

Where a is a number between 1 and 10. Technically, a is called the significand.

To convert a really big number from decimal notation to scientific notation, shift the decimal point to the left until you get a number between 1 and 10. Get rid of all the extra zeroes, and plug in that number as the significand. Then count the number of places you moved the decimal point. Plug in that number for b.

To convert a really small number, it's almost the same thing, but shift the decimal point to the right until you get a number between 1 and 10. Get rid of the extra zeroes and plug in that number as the significand. Then count the number of places you moved the decimal point, and plug in the opposite of that number for b.

To convert from scientific notation back to decimal notation, you just do the opposite. If the exponent is positive, move the decimal point that many places to the right, adding zeroes where necessary. If the exponent is negative, move the decimal point that many places to the left.

To add or subtract, just make sure the exponents on the 10 are the same for both numbers. Then add or subtract the two significands.

To multiply or divide, multiply or divide the significands. For the exponents, use the same rules that you'd use for any other exponents. For multiplication, you add the exponents; for division, you subtract them.

This makes a lot more sense once you start using it, so let's try some practice problems.

## Problem 1

We'll start off with a simple one, just to get you going. If you don't have anything handy, you might want to grab a pen and paper so you can do some scratch work.

The human body contains 1 x 10^14 cells, 1/10 of which are actually human cells and 9/10 of which are bacterial cells. Approximately how many bacterial cells are in the human body?

(A) 1 x 10^14

(B) 1 x 10^13

(C) 9 x 10^14

(D) 9 x 10^13

Let's go through this step by step. We know that the number of bacterial cells is 9/10, or 90%, of 1 x 10^14. We want to take 90% of 1 x 10^14, so we need to multiply by 0.9. First, we'll convert 0.9 to scientific notation:

9 x 10^-1

Next, we'll multiply.

(1 x 10^14) x (9 x 10^-1)

Multiply the significands to get 9. Then use exponent rules to deal with the exponents. When you multiply two exponential expressions with like bases, you add the exponents.

14 + -1 = 13

So our final answer is 9 x 10^13, or choice (D).

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