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ACT Prep: Practice & Study Guide43 chapters | 347 lessons

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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.

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.

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).

*Planet A is 3 x 10^14 light-years away from Planet B. Planet B is 2 x 10^12 light-years away from Planet C. If Spaceman Spiff flies from Planet A to Planet B to Planet C, how many light-years does he travel in total?*

*(A) 5 x 10^26*

*(B) 3.2 x 10^26*

*(C) 3.02 x 10^14*

*(D) 5 x 10^14*

In simple terms, this problem is:

*3 x 10^14 + 2 x 10^12*

You could do this by converting both numbers back to decimal notation, adding, and then converting back to scientific notation. This is unwieldy, though, and ties you up in a lot of zeroes, which is exactly what scientific notation was designed to avoid. Instead, we'll rewrite the two numbers to make the exponents the same, so we can add them in scientific form.

Remember that the exponent on the 10 represents the number of places that the decimal point has moved to the left. We can change this exponent by manipulating the position of the decimal point in the significand:

*2 x 10^12 = 0.2 x 10^13 = 0.02 x 10^14*

For each shift of the decimal point to the left, we simply add one to the exponent. This is exactly what we did when converting the number from decimal to scientific notation in the first place. Normally, you'd stop when the significand is between 1 and 10, because those numbers are easy for people to read and use, but there's no mathematical reason why you have to do this. It's just as correct to write *0.02 x 10^14*, and it lets us add the two numbers:

*(0.02 x 10^14) + (3 x 10^14) = 3.02 x 10^14*

So answer choice (C) is correct.

In this lesson, you got some practice with scientific notation. Scientific notation isn't anything fancy; it's just a system for abbreviating very large or very small numbers, so you don't have to write out a bunch of zeroes every time.

All numbers in scientific notation are expressed in the form:

*a x 10^b*

Scientific notation is one of those concepts that sounds a lot more complicated to explain than it actually is: the best way to learn it is just by doing it. So now go ahead and try some questions on your own in the quiz!

Upon completing this lesson, you should be able to:

- Define scientific notation
- Identify the form for scientific notation
- Explain how to convert numbers to scientific notation

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ACT Prep: Practice & Study Guide43 chapters | 347 lessons

- Arithmetic with Whole Numbers 9:43
- What are the Different Types of Numbers? 6:56
- What is a Decimal Place Value? 6:19
- Comparing and Ordering Decimals 8:56
- Arithmetic with Decimal Numbers 10:40
- How to Build and Reduce Fractions 3:55
- Scientific Notation: Definition and Examples 6:49
- Scientific Notation: Practice Problems 6:31
- How to Find the Prime Factorization of a Number 5:36
- How to Find the Least Common Multiple 5:37
- How to Add and Subtract Like Fractions and Mixed Numbers 4:14
- How to Add and Subtract Unlike Fractions and Mixed Numbers 6:46
- Multiplying Fractions and Mixed Numbers 7:23
- Dividing Fractions and Mixed Numbers 7:12
- Practice with Fraction and Mixed Number Arithmetic 7:50
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