MARKED (2009/06/07)

Exponents are a short form for repeated multiplication of the same number by itself. For example, the short form for multiplying three copies of the number 5 or (5)(5)(5) is 5 to the power of 3 or 5^3. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The number that is being multiplied, being 5 in this example, is called the "base".

Exponent Laws

There are three different exponent laws. These three laws only work if both numbers have the same base.

Law 1.
Multiplication Law - You must add
the exponents together when you multiply powers with the same base.
3^4 X 3^4 = 3^8
4^4 X 4^6=4^10
6^7 X 6^3=6^10
Remember that if the base has no exponent, it counts as to the power of 1.

3^2 X 3=3^3

Law 2.
Division Law - You must subtract the exponents from each other when you divide powers with the same base.
6^5 / 6^2 = 6^3
5^5 / 5^2=5^3
4^5 / 4^4=5

Law 3.
Power of a Power Law - when raising a power, you
multiply the exponents.
(3^2)^4 = 3^8

*Remember to always Keep The Base The Same*

Zero Exponents

When a number is raised to the exponent of zero, the answer will always be one.

Example #1
3^0 = 1
5^0 = 1
100,000,000,000^0 = 1
See It's alway
ONE when you have an exponent of ZERO!!!

To try and explain this, consider the following...
Any number divided by itself will equil 1, and if you use exponent laws, 5^5/5^5 is equil to 5^0.

Negative Exponents
When a number is raised to a negative exponent, you can re-write the power as a positive exponent. A negative exponent is the opposite of a positive exponent.

There are three steps to finding the answer of an expression with a negative exponent:

1. Flip the base over
2. Rewrite the exponent as a positive
3. Simplify

Another way to re-write a negative exponent as a positive exponent is to divide it by 1 and turn the exponent into a positive exponent
Here are a few examples.....

X^-3 = 1/X^3

5^-3 = 1/5^3

Fractions With Exponents
When a fraction is raised to an exponent, we must apply the exponent to the numerator and the denominator.
As well if the fraction is raised to a negative exponent we must follow the rules of negative exponents.

Example #1
Rewrite the following as a positive exponent.

a) 8 ^ -3 Here, you can see that your negative exponent is 3.
8 ^ -3 This is the equation in fraction form. You can think of it as 8^ -3 divided by 1. 1 = 1 Flip the equation. Making 8^ -3 the denominator, this makes it positive. 8^3 = 1 Simplifying this equation is simply calculating 8^3, which is 8x8x8 or 512. b) (3/5)^ -1 In this equation, your negative exponent is 1. = (5/3)^1 Flip the fraction. This is the same thing as taking the reciprocal of the base and raising it to positive 1. = 5/3 To simplify this is equation you need to calculate (5/3)^1, which is 5/3.

Scientific Notation
Scientific notation,is a number that has the form of aX10^b, where a number is greater than or equal to 1 but less than 10 and has no more than two decimal places, and 10^b is a power of 10.

To write a number in scientific notation...
You put the decimal after the first digits and drop the zero, if there are more numbers then round them.
Then count the number of numbers that there are after the decimal, and that will be your expoment.

Example #1.. 29,000,000,000

You move the decimal so that it is between a number that is bigger than 0 but smaller than 10.
So it will be 2.9 * 10^10


You could also use scientific notation with small numbers...


Example #2

You move the decimal right so that it would be between the five and eight

So it will turn out to be 5.8 * 10^-5
Multiplying Numbers in Scientific Notation
a) (5.6 * 10^7) X (4.5 * 10^4)
=25.2 * 10^11
=2.52 * 10^12

b) (3.4 * 10^-5) X (4.5 * 10^-3)
=15.3 * 10^-8
=1.53 * 10^-7



Exponent rules math learning upgrade (2008). Retrieved May 12, 2009, from
Where do you use exponents in everyday life (n.d.). Retrieved June 1, 2009, from
Stapel, E. (n.d.). "Fractional (rational) exponents." . Retrieved April 22, 2009,

ancourt (2008). Unit #1 -Number Sense:. Retrieved December 15, 2008

Hendriks, J Math Unit 1. Negative Exponents (Podcast) Retrieved December 15, 2008

Rancourt (2008, September 12). Unit #1 - Number Sense:. Retrieved December 17, 2008, from

Rancourt. Number Sense-Zero and Negative Exponents/Fractions with Exponents. n.d. 18 Dec. 2008.

Rancourt, Exponents and Scientific Notation

Kevin Le, Kevin B, Jared P.