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The Laws of Exponents
PLEASE NOTE THAT YOU CANNOT USE A CALCULATOR ON THE
ACCUPLACER 
ELEMENTARY ALGEBRA TEST ! YOU MUST BE ABLE TO DO THE FOLLOWING PROBLEMS
WITHOUT A CALCULATOR!
The Laws of Exponents
Please be aware that the letters a, b, m , and n are
replacements for any real number .
However, when the letters are identical, we must use the SAME number
replacement!
When an exponential expression is When an exponential expression is divided When a number is raised to a negative 
Any number, except for 0, raised to the zero When a product is raised to a power, each When an exponential expression is raised When a fraction is raised to a power, this 
Problem 1:
Multiply
Problem 2:
Multiply
Problem 3:
Multiply
NOTE:
Problem 4:
Multiply
NOTE:
Please note that the law tells us that we have to have
identical numbers
in the base before we can add the exponents.
Problem 5:
Multiply
When you are multiplying two or more terms containing
variables , the
operation becomes easier if you group together the numbers and the
exponential expressions with like base and like powers as follows:
NOTE: You do not have to write down the "grouping"
step . Instead you
can write the answer right away.
Problem 6:
Multiply
This multiplication becomes easier to group as follows:
Problem 7:
Multiply
Problem 8:
Divide
Problem 9:
Divide
Problem 10:
Divide
NOTE:
Problem 11:
Divide
NOTE:
Problem 12:
Divide
This immediately allows us to illustrate the law
, which states
that any number raised to the zero power results in a value of 1.
Therefore, . Please note that .
Problem 13:
Find the value of
Problem 14:
Find the value of
Problem 15:
Find the values of
Please note that
By the Order of Operation, exponential expressions are
simplified BEFORE we multiply
(in this case by 1)!
Problem 16:
Divide
This immediately allows us to illustrate the law
, where the
negative exponent indicates that the exponential expression is actually in the
denominator of a fraction with numerator 1. Then, when written as a fraction,
the negative sign in the exponent changes to a positive sign .
Therefore, . Please note that .
Problem 17:
Rewrite in terms of positive exponents:
Problem 18:
Rewrite in terms of positive exponents:
Problem 19:
Rewrite in terms of positive exponents:
Now observe,
Here we actually have and we have to go by the Order of Operation.
Problem 20:
Rewrite in terms of positive exponents:
Now observe,
Here we actually have and we have to go by the Order of Operation.
Problem 21:
Divide
Here we must group together the numbers and the
exponential expressions
with like base as follows:
NOTE: You do not have to write down the "grouping"
step. Instead you
can write the answer right away.
Problem 22:
Divide
Problem 23:
Divide . Write your answer with positive exponents only!
Problem 24:
Divide
Problem 25:
Simplify
The word "simplify" takes on many meanings in
mathematics. Often you must
figure out its meaning from the mathematical expression you are asked to
"simplify." Here we will be asked to "simplify" instead of finding the value of
the
number raised to the third power.
Problem 26:
Simplify
Problem 27:
Simplify
Problem 28:
Simplify
Note that the number 1 raised to any power will always have a value of 1.
Problem 29:
Simplify
Problem 30:
Simplify
Note that
Problem 31:
Simplify
Please note that Later
on
we will learn how to deal with sums and differences raised to a power!
Problem 32:
Simplify
Please note that this law extends to any product
containing infinitely many
factors.
Problem 33:
Simplify
then
and
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