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Properties of Exponents and Square Roots

9.1 Properties of exponents

We need to review some properties of exponents before we introduce
our new topic. We recall that by , we mean the product of 5 x's.
From this definition, we see that

,
,
and

From these examples we abstract the following properties of
exponents:

The equalities hold for any positive integers m and n and any numbers
x and y.

We also see from the definition that

and

So, we have in general that

if ,
and
if .

These equalities hold for any positive integers m and n and any
nonzero number x. These two properties are not esthetically pleasing
because they depend on the relative sizes of the exponents. We would
like to extend the definition of the exponents so that the property
holds regardless of the relative sizes of the exponents. Then, since

for any exponent m and any nonzero number x, we have to define to
mean 1 whenever x is not zero. Moreover, since

we have to define to mean . We summarize the new definitions:
for any nonzero number x.
for any positive integer n and any nonzero number x.
With these definitions, the property

holds for any positive integers m and n regardless of the relative sizes
of m and n and for all nonzero number x. In fact, it is not difficult to
show that the properties of exponents listed above hold for all
integers, including the negative exponents and for all nonzero
numbers x and y. We summarize the properties of exponents (also
called the laws of exponents":

(A)
(B)
(C)
(D)

These equalities hold for any integers m and n and for any nonzero
numbers x and y.

There are other properties of exponents, but since we do not use
them, we do not list them.

To get a concrete feeling for these properties, compute the following
using a scientific calculator:

(a) (a' )
(b) (b' )
(c) (c' )
(d) (d' )
(e) (e' )
(f) (f' )
(g) (g' )

Example:Simplify the following expressions:
(a)
(b)
(c)
(d)
Solution:
(a)
(b)
= =
Or by Property (C)
= by Property (B)
=
(c) or
(d)
=
=
=

Your instructor may specify the form of your answer, like
to leave the your final results all in positive exponents.
Follow your instructor's instruction.

Exercises 9.1

Simplify the following expressions: (Assume that all letters in the
expressions stand for some nonzero numbers.)

(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
(k)

9.2 Scientific notation
In studying sciences, particularly astronomy, we encounter very small
as well as very large numbers. Such numbers are usually represented
in so-called scientific notation. The scientific notation is a system of
writing a decimal number as the product of a number between 1 and
10 and a power of 10. For example, the number is written as , and
the number is written as . (That is, place the decimal point right
after the first nonzero digit and multiply the number by an
appropriate power of 10.) Recall that when we multiply a number by
a power of 10, we merely shift the decimal point. For example, when
we multiply the number by , we shift the decimal point of five
places to the right, so the number is . Also when we divide a number
by a power of 10, we shift the decimal point of the number to the left
by appropriate places. For example, means that we divide by , we
shift the decimal point of six places to the left, so that the decimal
representation of the number is .

When we multiply and using a scientific calculator, the calculator
displays the number in the scientific notation. However, the
calculator cannot display the number as . It displays the number in
some other way depending on the brand of calculator you are using.
You must learn to interpret the display of your calculator. You
cannot copy down the display on your calculator because it
means a totally different thing. We now give some examples of
computations involving scientific notation.

Example:
Convert the numbers into scientific notation and then
compute the expressions:

(a)
(b)
(c)
(d)

Solution:
(a)
=
=
=
(b)
=
=
=
(c)
=
=
=
(d)
=
=
=
=
We wish to make a few comments.

* The decimal part of the number in scientific notation is
seldom an exact number, and so the result of your
computations involving scientific notation should be
rounded off to a reasonable length. The equality
should be interpreted in the sense of approximate
equality.

* Your instructor may instruct you to use a scientific
calculator to do the computations once you convert the
numbers into scientific notation. In that case, you must
learn to enter the number in scientific notation into the
calculator and perform the computations. You must
pay close attention to your instructor's instruction.

Exercises 9.2
1. If the numbers are not already expressed in scientific notation,
express them in scientific notation and them compute the
expressions:
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
(k) (l)
(m) (n)

2. "one light-year" is the distance that the light travels in one year.
The speed of light is about meters per second.

(a) If 1 year consists of 365 days, approximately how many miles is 1
light-year? (1000 meters is equal to 0.621 mile.)

(b) If the distance from the Sun to the Earth is about meters, how
long will it take for the light to reach the Earth? Express your
answer in a reasonable unit.

3. The Earth goes around the Sun in approximately circular orbit
with the Sun as the center of the circle. The approximate distance
of the Earth from the Sun is meters.

(a) Compute the speed with which the Earth is going around the Sun.
Give the speed in meters per second, taking one year to be 365
days.

(b) Express the speed of the Earth in miles per hour if 1000 meters is
equal to 0.621 mile.

4. The radius of the Earth is about meters.

(a) How many miles is the radius of the Earth?

(b) What is the distance around the Earth (along the Equator)?

(c) If we travel along the Equator at the speed of 11 miles per hour,
how long will it take to go around the Earth? Express your answer
in a reasonable unit.

5. In describing the staggering amount of our national debt, which
was about 1.3 trillion dollars at the time, President Reagan said
that if it were possible to string out 10 dollars bills to the Moon,
1.3 trillion dollars worth of ten dollar bills would stretch to the
Moon and back. If the length of the ten dollar bill is 15.7
centimeters (or 0.157 meters) and the distance from the Earth to
the Moon is about meters, was it an accurate statement?

6. The thickness of a ten dollar bill is about meters. If 1.3 trillion
dollars worth of ten dollar bills were stacked up in one pile, how
high will the pile be? Express your answer in a reasonable unit.
(1000 meters is equal to 0.621 mile.)

9.3 Properties of square roots

The square root of a (positive) number x is a number whose square is
equal to the number x. The square root of x is denoted by the symbol
. Square roots of positive numbers are in general not so nice
numbers in the sense that they are unending decimal numbers.
However, there are "nice" square roots, and some of them are given
below:

Etc.
Before we state some properties of square roots, we would like to find
the following square roots using a calculator:
(a) (a' )
(b) (b' )
(c) (c' )
(d) (d' )

We find that (a) and (a' ) give the same number 3, and (b) and (b' )
give the same number . These examples illustrate the general
property that

(1)___and___for any positive number a.

We also find that (c) and (c' ) give the same number , and (d) and
(d' ) give the same number . Note that 7 is , and . These two
examples illustrate the property that

(2)___for all positive numbers a and b.

The property illustrated in (d) and (d' ) used to be important in the
olden days when square roots of numbers were found from a table.
(Inexpensive calculators became available only in early 1980's.) The
table usually listed approximate square roots of numbers between 1
and 100. So, if we wanted to find the (approximate) square of a
number like 500, we had to bring down the number under the square
root sign to below 100, like in the following:

The process of writing as is used to be called ( and is still called)
"simplification" of the radical expression. (The square root sign is
also called radical sign.) The number under the radical sign is called
radicand.

Example:
Simplify the following radical expressions, (assuming that
all letters stand for some positive numbers): (By the
statement, we mean to reduce the radicands to as small
numbers or expressions as possible.)
(a)
(b)
(c)
(d)
(e)

Solution:
(a)
(b)
(c)
(d)
(e) We first simplify as . Then,

Exercises 9.3
Simplify the following expressions: (Assume all letters stand for some
positive numbers.)
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
(k) (l)
(m) (n)
(o) (p)
(r) (s)
(t) (u)