**Using scientific notation
**

What is 5200000000000 x 760000000000000000000000000000?

You don't want to count all those zeroes. Wouldn't it be nice if I had told
you how many there were in the first place?

Scientific notation is a way to express large or small numbers without writing
long strings of zeros. You express them as base numbers times 'powers of
ten.'

What does 'powers of ten' mean?

We're used to using 'powers' in calculating areas. When you see an equation
containing something like r^{2}, you know it means r x r. This is called
'r squared' or 'r to the second power.'

10^{2}, then, would be ten squared, or 10 x 10, or 100.

10^{3} would be 10 x 10 x 10 or 1000.

10^{4} would be 10 x 10 x 10 x 10 or 10,000

You can see a pattern, right? 10^{35} would end up being 1 with 35 zeroes after it.

In the expression 10^{35}, 35 is the **exponent.** That
means the power to which 10 is raised. It also means the number of zeroes that
would be after the number, if you wanted to waste your time writing them all
out.

What about numbers that aren't 100, 1000, etc.? What about numbers like 300 and 6,000,000? Well, these are all just some number multiplied by some power of ten. That's how you would say them if you read them aloud; three hundred. Six million. That's how you write them, too.

300 = 3 x 100 = 3 x 10^{2}

6,000,000 = 6 x 10^{6}

56,000,000 =

73,598,000,000 =

Numbers smaller than 1 have negative exponents. For instance:

.1 = 1 x 10^{-1}

This means the same thing as 1/10.

.00012 = 12 x 10 ^{-5} (note the exponent is the number of places
you must move the decimal

point **right** to get the 12)

.0004 =

.00005693 =

For more practice with this skill, click here

On your calculator and in computer programs, scientific notation is written as ‘E’. The ‘E’ stands for ‘exponent.’ For instance:

If the calculator says 2.45E-15 it means 2.45 x 10^{-15}

6E7 means 6 x 10^{7}

Convert each of these calculator expressions to scientific notation:

62E9 =

5E-12 =

**Multiplying with scientific notation: **

To multiply powers of ten, you add their exponents together.

For example, you know that: 10,000 x 10 = 100,000.

In scientific notation, this would be: 10^{4} x 10^{1} = 10^{4+1} =
10^{5} .

For more practice with this skill, click here

To multiply two numbers in scientific notation, you multiply their base numbers and their powers of ten separately. Take a very simple example: 20 x 20 = 400

In scientific notation, this is: (2 x 10^{1}) x (2 x 10^{1})
= (2 x 2) x ( 10^{1} x 10^{1} )
= (4) x ( 10^{1+1} ) = 4x 10^{2} = 400

Estimate the answers for the following problems:

5.41x 10^{5} x 3.8 x 10^{7} =

72.1 x 10^{3} x 40x10^{6} =

Make up two problems of your own and estimate their solutions.

For more practice with this skill, click here

**Dividing with scientific notation: **

To divide powers of ten, you subtract their exponents.

For example, you know that: 10,000/ 10 = 1,000

In scientific notation, this would be: 10^{4} / 10^{1} = 10^{4
-1} = 10^{3}

For more practice with this skill, click here

To divide two numbers in scientific notation, you divide their base numbers and their powers of ten separately. Take a very simple example: 400/20 = 20

In scientific notation, this is: 4 x 10^{2} / 2 x 10^{1} =
(4/2) x ( 10^{2} / 10^{1} )
= (2) x ( 10^{2-1} ) = 2 x 10^{1} = 20

Estimate the answers for the following problems:

5.41 x 10^{5} / 3.8 x 10^{7} =

72.1 x 10^{3}
/ 40 x 10^{6} =

Make up two problems of your own and estimate their solutions:

Now - what is 5200000000000 x 760000000000000000000000000000?