Alphabetize Annotation and Powers of ten

10 to the Power 2

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

xii means 10 × x = 100

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

Example: 103 = 10 × 10 × 10 = 1,000

  • In words: x3 could exist called "10 to the third power", "10 to the power 3" or merely "10 cubed"

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

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

You tin multiply any number past itself as many times as you want using this notation (come across Exponents), but powers of 10 take a special apply ...

Powers of 10

"Powers of 10" is a very useful manner of writing downwards large or small numbers.

Instead of having lots of zeros, you testify how many powers of ten will brand that many zeros

Case: 5,000 = v × 1,000 = v × 10three

five grand is 5 times a k. And a thousand is 103. And so 5 times 10iii = 5,000

Tin yous see that 103 is a handy fashion of making 3 zeros?

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

Example: The Mass of the Sun

The Sunday has a Mass of 1.988 × 1030 kg.

It is too difficult to write 1,988,000,000,000,000,000,000,000,000,000 kg

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

Example: A Calorie-free Year (the distance light travels in 1 yr)

It is easier to employ 9.461 × 1015 meters, rather than ix,461,000,000,000,000 meters

It is unremarkably called Scientific Annotation, or Standard Form.

Other Fashion of Writing It

Sometimes people utilize the ^ symbol (to a higher place the half dozen on your keyboard), as information technology is easy to type.

Case: 3 × 10^4 is the same as three × ten4

  • 3 × x^four = three × 10 × 10 × 10 × 10 = xxx,000

calculator e notation

Calculators often utilise "Due east" or "due east" like this:

Example: 6E+ v is the aforementioned as 6 × 105

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

Example: 3.12E4 is the aforementioned as 3.12 × ten4

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

The Trick

While at first information technology may look hard, there is an like shooting fish in a barrel "trick":

The index of 10 says ...

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

Case: What is 1.35 × 104 ?

You can calculate it as: ane.35 x (10 × 10 × 10 × x) = 1.35 x 10,000 = xiii,500

But it is easier to recollect "move the decimal point 4 places to the right" like this:

Negative Powers of x

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

A negative ability means how many times to divide by the number.

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

Just 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.

Example: What is 7.ane × 10-3 ?

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

Merely information technology is easier to think "movement the decimal point 3 places to the left" like this:

Endeavor It Yourself

Enter a number and run across it in Scientific Notation:

Now try to use Scientific Notation yourself:

Summary

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

Number

In Scientific
Notation

In Words
Positive Powers five,000 5 × ten3 five K
Negative Powers 0.005 5 × 10-3 5 Gths