Study Guide
Topic: Introduction to Exponents, Product Rule of Exponents
1) Write the following in a “shorter/simpler way” using multiplication:
a.)
, b.)
, c.)
d.)
Remember: Multiplication is a “shorthand” for writing repeated addition of the same
number or object.
2.1) Alice is a strict Maths teacher. Whenever her students are late, she gives them a lot of
Maths exercises to do. One day, Bob was late and Alice got angry. Alice gave some tasks
for Bob to do. a.) Write “two times two” in Maths symbols., b.) Write “nine times nine
times nine” in Maths symbols. Pretend that you are Bob and do these tasks.
2.2) The next day, Bob was late again and now Alice is really angry. She is determined to
give Bob harder tasks. She commanded Bob, “Multiply four by itself 200 times. Write
these in Maths symbols on the blackboard.” Would you like to do this if you were Bob?
2.3) Luckily for Bob, a Maths Wizard came to his rescue. The wizard said, “Let us make a
notation to make multiplying same things easier to write. Let this notation be called
exponents which is to be written on the top-right portion of the number to be multiplied
by itself, which is to be called the base. The resulting number is then called a power of
that base to the certain exponent. This process is called exponentiation.” Bob thanked
the wizard and the wizard left. Bob wrote the down the answer to the task that Alice gave
her as
. In
, 4 is called the base, 200 is called the exponent, and
the 200th power of 4.
is called
can be read as “4 to the 200” or “4 to the 200th power”.
Remember: Exponentiation is a “shorthand” for writing repeated multiplication of the
same number or object.
3) Answer the following:
a.) Write down “12 to the 18”, b.) In
, which is the base and which is the exponent?
4) Write down the following using exponents:
a.)
, b.)
, c.)
, d.)
, e.)
, f.)
↻ Refresher: Recall that if there is an expression involving only the multiplication
operation, then it does not matter which terms are multiplied first. The answer will still
be the same. In other words, putting parenthesis grouping symbols will not change the
product. For example,
.
5.1) First, write down the expressions inside the parenthesis using exponents. Then,
remove all the parentheses to write an equivalent form of the whole expression.
a.)
d.)
, b.)
, c.)
5.2) Based on your observations in 5.1), a.) If
, what is x?, b.) What is
?,
c.) Given that
, what does this tell us about the exponents of the two numbers on
the left-hand side?, d.) Given a certain number a and positive integers m and n, what is
? This is called the Product Rule of Exponents.
5.3) Simplify the following expressions by writing as a single power:
a.)
, b.)
f.)
, c.)
, d.)
, e.)
,
(For a positive integer n,
. For example,
. n! is called the factorial of the integer n and can be read as “n
factorial”.)
WARNING! Be careful of interpreting expressions like
different expressions!
means
Author: cuvy_27_virl (Doc. # 1)
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while
means
and
. They are
.
Answer Key
1)
a.)
, b.)
, c.)
, d.)
2.1)
2.2)
a.)
, b.)
If only the standard multiplication symbol is to be used for this task, I would not do
it if I were Bob and tell my teacher Alice that the task is impractical and
unreasonable.
3)
a.)
4)
a.)
5.1)
a.)
5.2)
a.)
, b.) base: 13, exponent: 23
, b.)
, c.)
, d.)
, e.)
, f.)
, b.)
, b.)
, c.)
, d.)
, c.) The two exponents should be equal and they should add up
to 2. Therefore, it must be that their exponent is 1. In other words,
the same as writing it as
. This tells us that if a number a has no written
exponent, then the exponent is 1. (
5.3)
a.)
, b.)
, c.)
, d.)
is also
, e.)
), d.)
, f.)