As you may already know, exponents are a representation of multiplying a number by itself. For example, 5^4 (read "five to the fourth power") equals 5 X 5 X 5 X 5, or 625. Easy stuff.

 

What you may not know is that exponents of the same number, or base can be combined. Whaaaaat?

 

Really, if you think about this in multiplication form, it makes more sense. So, let's say we have two exponents of base 2: 2^5 and 2^2. We want to multiply them together to get one exponent, and because they have the same base, we can do that. We can lay it out as:

 

2^5 X 2^2 = (2 X 2 X 2 X 2 X 2) X (2 X 2) = 2 X 2 X 2 X 2 X 2 X 2 X 2 = 2^7.

 

Keep in mind here that we can't add different exponents, even if they had the same base. 2^5 + 2^2 = 32 + 4 = 36, which is much smaller than 2^7 = 128. But, multiplication allows us to combine exponents while keeping them in exponential notation (i.e., writing 7^4 instead of 7 X 7 X 7 X 7, which I really don't feel like doing). To multiply exponents of the same base, you just add the exponents together. In other words:

 

n^a X n^b = n^(a+b)

 

Dividing does the opposite. if you're dividing exponents of the same base, you subtract the exponents:

 

n^a/n^b = n^(a-b)

 

These rules are really helpful because they allow you to combine exponents quickly, which you'll need when you start getting into more complicated equations. Let's do some examples.

 

Ex 1) 10^4 X 10^7

 

Tens are easy to multiply because all you have to do is write out the zeroes behind the one. But, humor me for a second, and you'll be glad you did.

 

10^4 X 10^7 = 10^(4+7) = 10^11 = 10,000,000,000.

 

BAM. No multiplication required.

 

Ex 2) x^14/x^28

 

x^14/x^28 = x^(14-28) = x^(-14)

 

Hold up. Negative exponents? What does this even mean?

 

Negative exponents are just another way of writing the reciprocal of an exponent. So, 2^(-3) = 1/(2^3) = 1/8

 

 

 

Now that you have the basics down, you can apply these to find answers to problems you couldn't find before. Click here to access the next lesson on solving problems with exponents and roots.