# 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.

(c) Express the speed of the Earth in feet per second if 1
mile is equal

to 5280 feet.

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)