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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
multiplied by another exponential
expression having the same base, the
powers are added .

When an exponential expression is divided
by another exponential expression having
the same base, the power in the
denominator is subtracted from the power in
the numerator.

When a number is raised to a negative
power, the exponential expression can be
placed in the denominator of a fraction with
numerator 1, but the negative sign in the
exponent changes to a positive sign.

Any number, except for 0, raised to the zero
power results in a value of 1.

When a product is raised to a power, each
factor is raised to the power.

When an exponential expression is raised
to a power, the powers are multiplied.

When a fraction is raised to a power, this
power can be distributed to the numerator
and to the denominator.

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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