Index Notation and Powers of 10

10 to the Power 2

The exponent (or index or power) of a number says
how many times to utilize the number in a multiplication.

102 means 10 × 10 = 100

(It says x is used two times in the multiplication)

Example: 103 = 10 × 10 × 10 = 1,000

  • In words: 103 could exist called "10 to the 3rd power", "10 to the ability iii" or simply "x cubed"

Case: ten4 = 10 × x × 10 × 10 = 10,000

  • In words: 104 could exist called "10 to the 4th power", "x to the power 4" or "10 to the 4"

You lot can multiply any number by itself as many times as you want using this note (see Exponents), but powers of x have a special utilize ...

Powers of 10

"Powers of 10" is a very useful style of writing down large or small numbers.

Instead of having lots of zeros, you lot show how many powers of ten will make that many zeros

Example: 5,000 = 5 × i,000 = five × 10iii

five chiliad is 5 times a yard. And a 1000 is 103. So v times x3 = 5,000

Can you lot see that 103 is a handy way of making iii zeros?

Scientists and Engineers (who oftentimes use very big or very small numbers) like to write numbers this way.

Case: The Mass of the Sunday

The Sun has a Mass of 1.988 × x30 kg.

Information technology is too hard to write 1,988,000,000,000,000,000,000,000,000,000 kg

(And very like shooting fish in a barrel to make a mistake counting the zeros!)

Example: A Light Year (the distance light travels in one twelvemonth)

It is easier to use ix.461 × 10xv meters, rather than nine,461,000,000,000,000 meters

Information technology is commonly chosen Scientific Notation, or Standard Form.

Other Way of Writing It

Sometimes people use the ^ symbol (above the 6 on your keyboard), as information technology is easy to blazon.

Case: iii × 10^iv is the same every bit 3 × ten4

  • 3 × 10^4 = 3 × 10 × 10 × 10 × x = xxx,000

calculator e notation

Calculators often apply "E" or "e" like this:

Example: 6E+ 5 is the same as 6 × 105

  • 6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000

Example: 3.12E4 is the same as 3.12 × 104

  • iii.12E4 = 3.12 × 10 × ten × 10 × 10 = 31,200

The Trick

While at first it may await hard, there is an like shooting fish in a barrel "fob":

The index of x says ...

... how many places to move the decimal point to the correct.

Example: What is i.35 × xfour ?

You lot can calculate it as: i.35 x (10 × x × 10 × 10) = one.35 x 10,000 = 13,500

Only it is easier to retrieve "move the decimal point 4 places to the right" like this:

Negative Powers of 10

Negative? What could be the opposite of multiplying? Dividing!

A negative power means how many times to dissever by the number.

Example: 5 × ten-3 = 5 ÷ 10 ÷ ten ÷ x = 0.005

But remember for negative powers of 10:

For negative powers of 10, move the decimal point to the left.

And then Negatives just go the other manner.

Case: What is vii.1 × 10-3 ?

Well, it is really 7.ane ten ( 1/10 × ane/10 × 1/10 ) = seven.ane × 0.001 = 0.0071

But it is easier to think "motility the decimal point 3 places to the left" similar this:

Effort It Yourself

Enter a number and run into information technology in Scientific Annotation:

Now endeavour to use Scientific Notation yourself:

Summary

The index of 10 says how many places to move the decimal bespeak. Positive means movement it to the right, negative means to the left. Example:

Number

In Scientific
Annotation

In Words
Positive Powers v,000 5 × 10three 5 1000
Negative Powers 0.005 5 × 10-3 5 Yardths