Exponents: The Basics

Why do we use exponents? When you have very large or very small numbers, it can be awkward to write them out. To save space (and headaches!), people use exponents.

What are exponents? Exponents are the numbers above and to the right of 10 in the following: 10³, 10¹⁰, 10^-3, etc. We call 10 the "base" to the exponents.

• The exponent indicated the number of times "10" is multiplied by itself. So, 10³= 10*10*10, or 1,000.
• We see that the exponent indicates the number of zeroes that follow the 1. (For 10³, which is 1,000, three zeros are after the 1.)

A negative exponent (10^-x) is equal to 1/10^x

• For example, 10^-3 = 1 x 10^-3 = 1/10³ = 1/1,000 = 0.001.

But, most quantities cannot simply be written as base 10 to some exponent. Let's look at some examples...

• What do we mean by the number 3 x 10⁸ m/s (this is approximately the speed of light)? 
  • This means 3 x 100,000,000 m/s  = 300,000,000 m/s (or three hundred million meters per second).
• It takes someone about 5 x 10^-2 seconds to blink their eyes.
  • This can be written as: 5 x 1/10² = 5/100 = 0.05 seconds (or five hundredths of a second).

Let's look at some Very LARGE numbers:

There are about 7,000,000,000,000,000,000,000,000,000 atoms in a human body. If you count the number of zeroes in this number, you should get 27.

• We can write this as 7 x 10²⁷. You can read this as "7 followed by 27 zeroes."

The mass of the Sun is about 2,000,000,000,000,000,000,000,000,000,000 kg. There are 30 zeroes in this number (10 sets of 3 zeroes).

• So, we can write this as 2 x 10³⁰ kg. You can read this as "2 followed by 30 zeroes."

The volume of the Sun is about 1,400,000,000,000,000,000 km³. 

• We can write this as 14 x 10¹⁷ km³. We read this as "14 followed by 17 zeroes." Or, we could write this as 1.4 x 10¹⁸ km³. (since we divided 14 by 10, we need to multiply 10¹⁷ by 10 to keep the number the same magnitude).

Now let's look at some Very small numbers:

For example, the mass of a proton is about 0.00000000000000000000000000167 kg. This is even more awkward to write! 

• If you moved the decimal place to the right 27 places (or multiply by 10²⁷), you would get 1.67. This means if you started with 1.67, you would need to move the decimal place to the left 27 places to get the number for the mass of a proton (or divide by 10²⁷). 
• Dividing by 10²⁷ is the same as multiplying by 10^-27
• So, this number for the mass of a proton can be written as: 1.67 x 10^-27 kg. 

(Click here for a review of how math with exponents works.)