Exponents describe how many times to use a number in a multiplication. The laws of exponents are based on three main ideas: 1) the exponent says how many times to multiply the base, 2) a negative exponent means to divide the base, and 3) a fractional exponent means to take a root of the base. Some key exponent rules are: am * an = am+n, am / an = am-n, (ab)n = an * bn, and an/b = (a/b)n. A zero exponent equals 1 for any base.

1. Laws of Exponents
Exponents are also called Powers or Indices
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
could be called "8 to the second power", "8
to the power 2" or simply "8 squared"
Try it yourself:
So an Exponent just saves you writing out lots of multiplies!
Example: a7
a7
= a × a × a × a × a × a × a = aaaaaaa
Notice how I just wrote the letters together to mean multiply? We will do that a
lot here.
Example: x6
= xxxxxx

2. The Key to the Laws
Writing all the letters down is the key to understanding the
Laws
Example: x2
x3
= (xx)(xxx) = xxxxx = x5
Which shows that x2
x3
= x5
, but more on that later!
So, when in doubt, just remember to write down all the letters (as many as the
exponent tells you to) and see if you can make sense of it.
All you need to know ...
The "Laws of Exponents" (also called "Rules of Exponents") come from three
ideas:
The exponent says how many times to use the number in a
multiplication.
A negative exponent means divide, because the opposite of
multiplying is dividing
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.

3. 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
xm
xn
= xm+n
x2
x3
= x2+3
= x5
xm
/xn
= xm-n
x6
/x2
= x6-2
= x4
(xm
)n
= xmn
(x2
)3
= x2×3
= x6
(xy)n
= xn
yn
(xy)3
= x3
y3
(x/y)n
= xn
/yn
(x/y)2
= x2
/ y2
x-n
= 1/xn
x-3
= 1/x3
And the law about Fractional Exponents:
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: Powers of 5
.. etc..
52
1 × 5 × 5 25
51
1 × 5 5

4. 50
1 1
5-1
1 ÷ 5 0.2
5-2
1 ÷ 5 ÷ 5 0.04
.. etc..
Look at that table for a while ... notice that positive, zero or negative exponents
are really part of the same pattern, i.e. 5 times larger (or 5 times smaller)
depending on whether the exponent gets larger (or smaller).
The law that xm
xn
= xm+n
With xm
xn
, 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: x2
x3
= (xx)(xxx) = xxxxx = x5
So, x2
x3
= 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
/x2
= (xxxx) / (xx) = xx = x2
So, x4
/x2
= x(4-2)
= 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 :

5. Example: x2
/x2
= x2-2
= x0
=1
The law that (xm
)n
= xmn
First you multiply "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
= xn
yn
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) = x3
y3
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
OK, this one is a little more complicated!

6. I suggest you read Fractional Exponents first, or this may not make
sense.
Anyway, the important idea is that:
x1/n
= The n-th Root of x
And so a fractional exponent like 43/2
is really saying to do a cube (3)
and a square root(1/2), in any order.
Just remember from fractions that m/n = m × (1/n):
Example:
The order does not matter, so it also works for m/n = (1/n) × m:
Example:
And That Is It!
If you find it hard to remember all these rules, then remember this:
you can work them out when 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

7. Negative Exponent (n<0) Undefined! (Because dividing by 0 is undefined)
Exponent = 0 Ummm ... see below!
The Strange Case of 00
There are two different arguments for the correct value of 00
.
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"
Exponent rules
Exponent rules, laws of exponent and examples.
 What is an exponent
 Exponents rules
 Exponents calculator
What is an exponent
The base a is raised to the power of n is equal to the multiplication of a, n times:
an
= a × a × ... × a
n times
a is the base and n is the exponent.
Examples
31
= 3
32
= 3 × 3 = 9
33
= 3 × 3 × 3 = 27
34
= 3 × 3 × 3 × 3 = 81
35
= 3 × 3 × 3 × 3 × 3 = 243
Exponents rules and properties

8. Rule name Rule Example
Product rules
an
· am
= an+m
23
· 24
= 23+4
= 128
an · bn = (a · b)n 32 · 42 = (3·4)2 = 144
Quotient rules
an / am = an-m 25 / 23 = 25-3 = 4
an / bn = (a / b)n 43 / 23 = (4/2)3 = 8
Power rules
(bn
)m
= bn·m
(23
)2
= 23·2
= 64
bnm
= b(nm
) 232
= 2(32
)= 512
m√(bn) = b n/m 2√(26) = 26/2 = 8
b1/n = n√b 81/3 = 3√8 = 2
Negative exponents b-n = 1 / bn 2-3 = 1/23 = 0.125
Zero rules
b0
= 1 50
= 1
0n = 0 , for n>0 05 = 0
One rules
b1 = b 51 = 5
1n = 1 15 = 1
Minus one rule (-1)5 = -1
Derivative rule (xn)' = n·x n-1 (x3)' = 3·x3-1
Integral rule ∫ xndx = xn+1/(n+1)+C ∫ x2dx = x2+1/(2+1)+C
Exponents product rules
Product rule with same base
an
· am
= an+m

9. Example:
23
· 24
= 23+4
= 27
= 2·2·2·2·2·2·2 = 128
Product rule with same exponent
an
· bn
= (a · b)n
Example:
32
· 42
= (3·4)2
= 122
= 12·12 = 144
See: Multplying exponents
Exponents quotient rules
Quotient rule with same base
an
/ am
= an-m
Example:
25
/ 23
= 25-3
= 22
= 2·2 = 4
Quotient rule with same exponent
an
/ bn
= (a / b)n
Example:
43
/ 23
= (4/2)3
= 23
= 2·2·2 = 8
See: Dividing exponents
Exponents power rules
Power rule I
(an
) m
= a n·m
Example:
(23
)2
= 23·2
= 26
= 2·2·2·2·2·2 = 64
Power rule II
a nm
= a (nm
)
Example:
232
= 2(32
) = 2(3·3) = 29 = 2·2·2·2·2·2·2·2·2 = 512
Power rule with radicals
m
√(a n
) = a n/m
Example:
2
√(26
) = 26/2
= 23
= 2·2·2 = 8

10. Negative exponents rule
b-n
= 1 / bn
Example:
2-3
= 1/23
= 1/(2·2·2) = 1/8 = 0.125
Negative exponents
How to calculate negative exponents.
 Negative exponents rule
 Negative exponent example
 Negative fractional exponents
 Fractions with negative exponents
 Multiplying negative exponents
 Dividing negative exponents
Negative exponents rule
The base b raised to the power of minus n is equal to 1 divided by the base b raised to the power of
n:
b-n
= 1 / bn
Negative exponent example
The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of
3:
2-3
= 1/23
= 1/(2·2·2) = 1/8 = 0.125
Negative fractional exponents
The base b raised to the power of minus n/m is equal to 1 divided by the base b raised to the power
of n/m:
b-n/m
= 1 / bn/m
= 1 / (m
√b)n
The base 2 raised to the power of minus 1/2 is equal to 1 divided by the base 2 raised to the power
of 1/2:
2-1/2
= 1/21/2
= 1/√2 = 0.7071
Fractions with negative exponents
The base a/b raised to the power of minus n is equal to 1 divided by the base a/b raised to the power
of n:
(a/b)-n
= 1 / (a/b)n
= 1 / (an
/bn
) = bn
/an

11. The base 2 raised to the power of minus 3 is equal to 1 divided by the base 2 raised to the power of
3:
(2/3)-2
= 1 / (2/3)2
= 1 / (22
/32
) = 32
/22
= 9/4 = 2.25
Multiplying negative exponents
For exponents with the same base, we can add the exponents:
a -n
· a -m
= a -(n+m)
= 1 / a n+m
Example:
2-3
· 2-4
= 2-(3+4)
= 2-7
= 1 / 27
= 1 / (2·2·2·2·2·2·2) = 1 / 128 =
0.0078125
When the bases are diffenrent and the exponents of a and b are the same, we can multiply a and b
first:
a -n
· b -n
= (a · b) -n
Example:
3-2
· 4-2
= (3·4)-2
= 12-2
= 1 / 122
= 1 / (12·12) = 1 / 144 =
0.0069444
When the bases and the exponents are different we have to calculate each exponent and then
multiply:
a -n
· b -m
Example:
3-2
· 4-3
= (1/9) · (1/64) = 1 / 576 = 0.0017361
Dividing negative exponents
For exponents with the same base, we should subtract the exponents:
a n
/ a m
= a n-m
Example:
26
/ 23
= 26-3
= 23
= 2·2·2 = 8
When the bases are diffenrent and the exponents of a and b are the same, we can divide a and b
first:
a n
/ b n
= (a / b) n
Example:
63
/ 23
= (6/2)3
= 33
= 3·3·3 = 27

12. When the bases and the exponents are different we have to calculate each exponent and then
divide:
a n
/ b m
Example:
62
/ 33
= 36 / 27 = 1.333
Zero exponents
Zero exponent rule and examples.
 Zero exponents rule
 Zero exponents examples
Zero exponents rule
The base b raised to the power of zero is equal to one:
b0
= 1
Zero exponents examples
Five raised to the power of zero is equal to one:
50
= 1
Minus five raised to the power of zero is equal to one:
(-5)0
= 1
Zero to raised the power of zero is equal to one:
00
= 1