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Laws of Exponents

Exponents are also called Powers or Indices

8 to the Power 2

The exponent of a number says how many times to multiply the number.

In this example: 82 = 8 × 8 = 64

  • In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared"

All you need to know ...

The "Laws of Exponents" (also called "Rules of Exponents"), all come from three ideas:

The exponent of a number says to multiply the number by itself so many times
   
The opposite of multiplying is dividing, so a negative exponent means divide
   
A fractional exponent like 1/n means to take the nth root:

If you understand those, then you understand exponents!

And all the laws below are based on those ideas.

Laws of Exponents

Here are the Laws (explanations follow):

Law Example
x1 = x 61 = 6
x0 = 1 70 = 1
x-1 = 1/x 4-1 = 1/4
xmxn = xm+n x2x3 = x2+3 = x5
xm/xn = xm-n x4/x2 = x4-2 = x2
(xm)n = xmn (x2)3 = x2×3 = x6
(xy)n = xnyn (xy)3 = x3y3
(x/y)n = xn/yn (x/y)2 = x2 / y2
x-n = 1/xn x-3 = 1/x3

Laws Explained

The first three laws above (x1 = x, x0 = 1 and x-1 = 1/x) are just part of the natural sequence of exponents. Have a look at this example:

Example: Powers of 5
  .. etc..  
52 1 × 5 × 5 25
51 1 × 5 5
50 1 1
5-1 1 ÷ 5 0.2
5-2 1 ÷ 5 ÷ 5 0.04
  .. etc..  

You will see that positive, zero or negative exponents are really part of the same pattern, i.e. 5 times larger (or smaller) depending on whether the exponent gets larger (or smaller).

The law that xmxn = xm+n

With xmxn, how many times will you end up multiplying "x"? Answer: first "m" times, then by another "n" times, for a total of "m+n" times.

Example: x2x3 = (xx) × (xxx) = xxxxx = x5

So, x2x3 = x(2+3) = x5

The law that xm/xn = xm-n

Like the previous example, how many times will you end up multiplying "x"? Answer: "m" times, then reduce that by "n" times (because you are dividing), for a total of "m-n" times.

Example: x4-2 = x4/x2 = (xxxx) / (xx) = xx = x2

(Remember that x/x = 1, so every time you see an x "above the line" and one "below the line" you can cancel them out.)

This law can also show you why x0=1 :

Example: x2/x2 = x2-2 = x0 =1

The law that (xm)n = xmn

First you multiply x "m" times. Then you have to do that "n" times, for a total of m×n times.

Example: (x3)4 = (xxx)4 = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x12

So (x3)4 = x3×4 = x12

The law that (xy)n = xnyn

To show how this one works, just think of re-arranging all the "x"s and "y" as in this example:

Example: (xy)3 = (xy)(xy)(xy) = xyxyxy = xxxyyy = (xxx)(yyy) = x3y3

The law that (x/y)n = xn/yn

Similar to the previous example, just re-arrange the "x"s and "y"s

Example: (x/y)3 = (x/y)(x/y)(x/y) = (xxx)/(yyy) = x3/y3

The law that

To understand this, just remember from fractions that n/m = n × (1/m):

Example:

And That Is It

If you find it hard to remember all these rules, then remember this:
you can always work them out if you understand the three ideas at the top of this page.

Oh, One More Thing ... What if x= 0?

Positive Exponent (n>0) 0n = 0
Negative Exponent (n<0) Undefined! (Because dividing by 0)
Exponent = 0 Ummm ... see below!

The Strange Case of 00

There are two different arguments for the correct value. 00 could be 1, or possibly 0, so some people say it is really "indeterminate":
x0 = 1, so ... 00 = 1
0n = 0, so ... 00 = 0
When in doubt ... 00 = "indeterminate"