We already realised with law 3 that a minus sign is another way of saying that the exponential number is to be divided instead of multiplied. Law 4 is just a more general way of saying the same thing. We get this law by multiplying law 3 by
on both sides and using law 2.
For example,
Application using exponential law 4:
Exponential law 5:
The order in which two real numbers are multiplied together does not matter. Therefore,
For example,
Application using exponential law 5:
Exponential law 6:
We can find the exponential of an exponential of a number. An exponential of a number is just a real number. So, even though the sentence sounds complicated, it is just saying that you can find the exponential of a number and then take the exponential of that number. You just take the exponential twice, using the answer of the first exponential as the argument for the second one.
For example,
Application using exponential law 6:
Simplify:
Investigation : exponential numbers
Match the answers to the questions, by filling in the correct answer into the
Answer column.
Possible answers are:
, 1,
,
, 8. Answers may be repeated.
Question
Answer
The following video gives an example on using some of the concepts covered in this chapter.
Summary
Exponential notation means a number written like
where
is an integer and
can be any real number.
is called the
base and
is called the
exponent or
index .
The
th power of
is defined as:
There are six laws of exponents:
Exponential Law 1:
Exponential Law 2:
Exponential Law 3:
Exponential Law 4:
Exponential Law 5:
Exponential Law 6:
End of chapter exercises
Simplify as far as possible:
Simplify without using a calculator. Leave your answers with positive exponents.