Index Notation and Powers of 10

(Note: Index and Power mean the same things as Exponent)

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

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

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

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

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 would be 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

  • 3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

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:

1.35 13.5 135. 1350. 13500.

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:

7.1 0.71 0.071 0.0071

Try it Yourself!

Just type in a number here:

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 Powers 5,000 5 × 103 5 Thousand
Negative Powers 0.005 5 × 10-3 5 Thousandths