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 use the number in a multiplication.

102 means 10 × 10 = 100

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

Example: 103 = 10 × 10 × 10 = 1,000

Example: 104 = 10 × 10 × 10 × 10 = 10,000

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

Powers of 10

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

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

Example: 5,000 = 5 × 1,000 = 5 × 103

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

Can you see that 103 is a handy way of making 3 zeros?

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

Example: The Mass of the Sun

The Sun has a Mass of 1.988 × 1030 kg.

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

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

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

It is easier to use 9.461 × 1015 meters, rather than 9,461,000,000,000,000 meters

It is commonly called Scientific Notation, or Standard Form.

Other Way of Writing It

Sometimes people use the ^ symbol (above the 6 on your keyboard), as it is easy to type.

Example: 3 × 10^4 is the same as 3 × 104

calculator e notation

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

Example: 6E+5 is the same as 6 × 105

Example: 3.12E4 is the same as 3.12 × 104

The Trick

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

The index of 10 says ...

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

 

Example: What is 1.35 × 104 ?

You can calculate it as: 1.35 x (10 × 10 × 10 × 10) = 1.35 x 10,000 = 13,500

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

1.35arrow13.5arrow135.arrow1350.arrow13500.

Negative Powers of 10

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

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

Example: 5 × 10-3 = 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.

So Negatives just go the other way.

Example: What is 7.1 × 10-3 ?

Well, it is really 7.1 x (1/10 × 1/10 × 1/10) = 7.1 × 0.001 = 0.0071

But it is easier to think "move the decimal point 3 places to the left" like this:

7.1arrow0.71arrow0.071arrow0.0071

Try It Yourself

Enter a number and see it in Scientific Notation:

Now try to use Scientific Notation yourself:

Summary

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

Number
In Scientific
Notation
In Words
Positive Powers5,0005 × 1035 Thousand
Negative Powers0.0055 × 10-35 Thousandths