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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 Prev Next