Index Annotation and Powers of ten

10 to the Power 2

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

x2 ways 10 × 10 = 100

(It says 10 is used 2 times in the multiplication)

Instance: x3 = 10 × ten × 10 = 1,000

  • In words: 103 could be called "x to the third power", "ten to the power 3" or simply "10 cubed"

Example: 10iv = 10 × ten × 10 × 10 = ten,000

  • In words: 104 could be called "10 to the fourth power", "ten to the power 4" or "ten to the four"

You can multiply any number by itself as many times equally you want using this notation (see Exponents), but powers of 10 have a special use ...

Powers of ten

"Powers of x" is a very useful way of writing down large or small numbers.

Instead of having lots of zeros, you bear witness how many powers of 10 will make that many zeros

Instance: v,000 = 5 × 1,000 = 5 × 10iii

5 thousand is 5 times a thousand. And a thousand is xthree. So 5 times 103 = 5,000

Can y'all come across that 103 is a handy way of making 3 zeros?

Scientists and Engineers (who often utilise very big or very small numbers) like to write numbers this fashion.

Example: The Mass of the Sun

The Sun has a Mass of 1.988 × x30 kg.

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

(And very easy to make a mistake counting the zeros!)

Instance: A Light Yr (the distance calorie-free travels in one year)

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

Information technology is normally called Scientific Annotation, or Standard Form.

Other Way of Writing It

Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is like shooting fish in a barrel to type.

Case: 3 × 10^iv is the aforementioned equally three × x4

  • three × ten^iv = iii × ten × 10 × x × 10 = 30,000

calculator e notation

Calculators ofttimes use "Due east" or "e" similar this:

Instance: 6E+ 5 is the aforementioned equally 6 × 10v

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

Example: 3.12E4 is the aforementioned as three.12 × xiv

  • 3.12E4 = three.12 × 10 × x × 10 × 10 = 31,200

The Play a trick on

While at first it may look hard, there is an easy "play tricks":

The index of 10 says ...

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

Example: What is 1.35 × 10four ?

You can summate information technology equally: ane.35 10 (10 × 10 × 10 × ten) = one.35 x ten,000 = xiii,500

Only it is easier to retrieve "move the decimal signal iv places to the correct" similar this:

Negative Powers of 10

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

A negative power ways how many times to divide by the number.

Case: v × 10-3 = 5 ÷ x ÷ 10 ÷ 10 = 0.005

Simply remember for negative powers of 10:

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

Then Negatives just go the other mode.

Instance: What is 7.i × x-iii ?

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

But it is easier to retrieve "move the decimal betoken three places to the left" similar this:

Try It Yourself

Enter a number and come across it in Scientific Note:

Now try to apply Scientific Notation yourself:

Summary

The alphabetize of 10 says how many places to move the decimal point. Positive means move it to the right, negative ways to the left. Example:

Number

In Scientific
Annotation

In Words
Positive Powers 5,000 5 × 103 5 Yard
Negative Powers 0.005 5 × x-iii 5 Thousandths