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# Laws of Exponents

Exponents are also called Powers or Indices

 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 :

## 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:

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

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

## The law that

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

## 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"