What does 10 to the power of negative 3 mean?

Exponents are also called Powers or Indices

Let us first look at what an "exponent" is:

	The exponent of a number says how many times to use the number in a multiplication. In this example: 82 = 8 × 8 = 64
In words: 82 can be called "8 to the second power", "8 to the power 2" or simply "8 squared"

Example: 53 = 5 × 5 × 5 = 125

In words: 53 can be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

In general:

an tells you to use a in a multiplication n times:

But those are positive exponents, what about something like:

8-2

That exponent is negative ... what does it mean?

Negative Exponents

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

Dividing is the inverse (opposite) of Multiplying.

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

Example: 8-1 = 1 ÷ 8 = 1/8 = 0.125

Or many divides:

Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008

But that can be done an easier way:

5-3 could also be calculated like:

1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008

That last example showed an easier way to handle negative exponents:

• Calculate the positive exponent (an)
• Then take the Reciprocal (i.e. 1/an)

To change the sign (plus to minus, or minus to plus) of the exponent,
use the Reciprocal (i.e. 1/an)

So, what about 8-2 ?

Example: 8-2 = 1 ÷ 8 ÷ 8 = 1/82 = 1/64 = 0.015625

More Examples:

It All Makes Sense

My favorite method is to start with "1" and then multiply or divide as many times as the exponent says, then you will get the right answer, for example:

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern.

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 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

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

Calculators often use "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

• 3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200

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:

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:

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:

What does to the power of negative 3 mean?

What does 10 to the power of 3 mean?

What does 10 to the negative power mean?

What does to the power of a negative number mean?