Module 1 Integral Exponents (1)

8/18/2019 Module 1 Integral Exponents (1)

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Y

X

(Effective Alternative Secondary Education)

MATHEMATICS II

MODULE 1

Integral Exponents

BUREAU OF SECONDARY EDUCATION

Department of Education

DepEd Complex, eralco A!enue, "a#i$ Cit%


2/23

Module 1Integral Exponents

What this module is about

This module is about algebraic exression !ith ositive" #ero and negativeexonents$ Here you !ill develo s%ills in re!riting algebraic exressions !ith#ero and negative exonents and learn to aly this in solving roblems$

What you are expected to learn

This module is designed for you to demonstrate understanding ofexressions !ith ositive" negative and #ero exonents" and&

'$ evaluate exressions involving integral exonents"

$ re!rite algebraic exressions !ith #ero and negative exonents"

$ solve exonential e*uations" and

+$ solve roblems involving exressions !ith exonents

How much do you know

A$ Simlify&

'$ , -$ (x)+

$ (.) /$ .ab , a.b

$ + , + , , 0$ (ab)(bc)(abc)

+$&

'

((

1$()

*+,

&(,

y x y x

.$-

)

.

.

'2$()

*+,

&

(,

y x

y x

(


3/23

3$ Evaluate the follo!ing exressions

'$ x2 -$ 4'b2

$ '222 /$ (5)()2

$ .x2 0$

,

ba

+$ (.)2 1$ (xy)2

.$

,

,

)b

a+

'2$ x2

C$ 6rite the follo!ing exressions !ith the negative exonents

'$ 5' -$-

-

−

−

y

x

$ '25' /$ (xy)5+

$ 05 0$ (+5')5

+$ '5+ 1$

.−

y

x

.$ x5y '2$

( )(

-.-

−

−

b

ba

7$ Solve for x in the follo!ing e*uations$

'$ +x 8 -+$ '2x 8 '222$ x 8 /+$ x 8 -+.$ x5' 8 +

Solve the follo!ing roblems involving exonents$

-$ The seed of sound is about .$' x '2 er second$ 9ind the distancetraveled by sound in one hour$

/$ After -+ days an amoeba !ill have aroximately reroduced '$0+. x'2'1 amoebas$ Exress the number of amoebas in standard form$

0$ The seed of light is '$0- x '2. miles er second$ 9ind the seed of light in %m:sec$ ('%m 8 $- mi)

.


4/23

1$ ;ne mole of hydrogen molecules has a mass of $2'- g and contains-$2 x '2 hydrogen molecules$ 6hat is the hydrogen moleculescontent of a '2 mole<

'2$=hysicist measuring the *uantity of electric charge in coulombs(c)found that one coulomb is e*ual to -$+ x '2'0 electrons$ Ho! manyelectrons are there in '2 coulombs<

What you will do

>esson '

Evaluation of Exressions !ith Integral Exonents

9or all real numbers a" and integers m and n" the follo!ing la!s of exonent alies to exressions having the same base&

a$ Multilication&am , an 8 am ? n

b$ =o!ers of =o!er (am)n 8 amn$

Examples:

+/ , 8 ? Add the exonents 8 .

0 ,,,, . in factored form8 +

$ () 8 () @et the roduct of the exonents0 -

8 ,,,,, - in factored form8 '-

$ a+ , a5 8 a+ ? (5) Add exonents8 a' or a

+$ * , * 8 -?*'? Add exonents of the same base

8 - .*

-


5/23

.$ (a.b) 8 a'2b- @et the roduct of the exonents

c$ 7ivision&am an 8 am 5 n

Examples:

'$ /. / 8 /. 4 Subtract the exonent8 / Exress as factors8 /,/8 +1 The *uotient

$ 'a- +a5 8 a- 4 (5) 7ivide the numerical coefficients and8 a1 subtract the exonents$

$ x

.

y

+

x

y 8x

. 4

y

+ 4 '

Subtract the exonents of 8 xy exressions having the same base$

Try this out

A$ Evaluate the follo!ing&

'$ + , + -$ ()

$ . , . /$ () , +

$ + , , , + 0$ () , ()

+$ '2

, '2

1$ (+ , )

.$ (-)(-)(-) '2$ '2 , .

3$ Simlify&

'$ (0) -$ (b-).

$ (/5')5 /$ (b)

$ (+) 0$ (.x.y)

+$ ( , ) 1$ (+c.de)

.$ (.+ , .5) '2$ (abc )

C$ 9ind the *uotient&

'$ '2. '2 -$ r / r -

$ 1 1 /$ '+a'2 a5.

$ . .5 0$ '0x-y xy+$ /5' /5 1$ '.xy+ .xy.$ +. + '2$ +a/b. c +abc

&


6/23

>esson

Bero and egative Exonents

Bero Exonents&

In the *uotient rule" to divide exressions !ith the same base" %ee thebase and subtract the exonents$

That is

nm

n

m

aa

a −=

o!" suose that you allo! m to e*ual n$ Then you have"

,aaaa mmm

m

== −

3ut you %no! that it is also true that

+=m

m

a

a

If you comare e*uations (') and ()" you can see that the follo!ingdefinition is reasonable$

Examples:

Dse the above definition to simlify each exression$

'$ '/2 8 ' any number !hose exonent is 2 is e*ual to '$

$ -x2 8 -(') 8 - In -x2" the exonent 2 alies only to x$

$ (ab)2 8 (')(') 8 ' 7istributive roerty

+$ 4y2 8 5(y2)8 5(') 8 5 In 5y2" the exonent 2 alies only to y$

E uation '

E uation

The Bero Exonent

9or any real number a !here a ≠ 2" a2 8 '

)


7/23

egative Exonents&

In the roduct rule" to multily exressions !ith the same base" %ee thebase and add the exonents$

That is" am

• an

8 am?n

o!" suose that you allo! one of the exonents to be negative andaly the roduct rule" then you have" suose that m 8 and n 8 5$

Then" am • an 8 a • a5 8 a ? (5) 8 a2 8 '

So" a • a5 8 ' 7ivide both sides by a"

6e get .. +

aa =

−

Therefore !e have this definition

Examples:

Simlify the follo!ing exressions$

(ote& To simlify !ill mean to !rite the exression !ith ositive exonents)

'$ y5.

8

&

+

y you get the recirocal

$ +5 8+)

+

1-12-2

+

-

+&

== you get the recirocal and simlify

$ (5)5 8 (*

+

1.12.12.2

+

1.2

+.

=−−−

=− you get the recirocal and simlify

egative Exonents&

9or any non#ero real number a and !hole number n"

n

n

aa

+=−

and a5n is the multilicative inverse: recirocal of an$

*


8/23

+$

(*

'

+

.

(

.

(

.

(

+

.

(

+

.

(.

.

=

=

=

−

'

(*

'

(*+

(*

'+ =•=÷

.$ x5 8..

(+(

x x=•

Caution& The exression +!5 and (+!)5 are not the same$ 7o you see !hy<

-$ +!5 8(( -+- ww =•

/$ (+!)5 8((

+)

+

1-12-2

+

1-2

+

wwww==

Suose that a variable !ith a negative exonent aears in thedenominator of an exression$

0$(

(

+

++

aa

=−

((

++ a

a=•

Try this out

A$ Simlify each exression$

'$ .2 -$ x2y

$ (m+n+)2 /$ (xy#)2

$ 0m2 0$ '2222

+$ 4/t2 1$ (ab)2

To divide fractions" get therecirocal and roceed tomultilication$

@et the recirocal to have aositive exonent" simlify then

The exonent 4 alies only tothe variable x" and not to thecoefficient $

To divide" get the recirocal of(

+

a and multi l $

Change the negative exonent in

the denominator to ositive

'


9/23

.$ *2 '2$ 4.a2

3$ Simlify each of the follo!ing exressions$

'$ a5'2 -$ '2x5.

$ 5+ /$ (y)5+

$ (5+)5 0$ 4.t5

+$

(

(

& −

1$

−-

+

x

.$ !5+ '2$

−(.

(

a

>esson

Solving Exonential E*uations

Exonential E*uations are e*uations such that the un%no!n is anexonent$ Solutions may be rational or irrational$ The method of getting thesolution is to e*uate the exonents of numbers !hich may have the same base$

Examples:

Solve for x in the follo!ing e*uations&

'$ x 8 0'

x 8 +

x 8 +

$ .x 8 '. Exress '. as a o!er of ." '. 8 . • . • .

.x 8 . E*uate the exonents and cancel the base

x 8 Therefore" x 8 $

Exress the right5hand member as a o!er of " 0'

8 • • • So" you have both sides as a o!er of the base$

Cancel the base" e*uate the exonents" therefore x8 +/

3


10/23

$ +x ? 8 ++ 3oth sides have the same base$

x ? + 8 + Cancel the base" e*uate the exonents$

x 8 Simlify

x 8 ' Therefore" x 8 '$

+$ 1x ? ' 8 /x Exress both sides as a o!er of " 1 8 • and

(x?') 8 (x) / 8 • •$

x ? 8 x 7istributive =roerty of Multilication$

x ? 8 x Simlify

5x 8 5

x 8 Therefore" x 8 $

Try this out

A$ Solve for x in the follo!ing e*uations$

'$ .x 8 -. -$ x 4 ' 8 +

$ 0x 8 /$ x ? 8

$ +x 8 )-

+

0$ x 4 8 -+

+$ '2x 8 '2222 1$ .x 4 8 '.

.$ x 8 -+ '2$ x ? 4 8 -

+

3$ Solve for x$

'$ /x 8 +1 -$ '2x ? . 8 '.

$ +5x 8 '- /$ /x 4 ' 8 /'

$ 0x 4 ' 8 02 0$ 5x 8 (*

+

+$ -x 4 + 8 -'- 1$ '-x 4 ' 8 +x 4 +

+


11/23

.$ +x ? '- 8'-+ '2$ x 4 4 8 .-

>esson +

Solving =roblems Involving Exressions !ith Exonents

Scientific otation&

It is not uncommon in scientific alications of algebra to find yourself !or%ing !ith very large or very small numbers$ Even in the time of Archimedes(0/ 4 ' 3$C$)" The study of such numbers !as not unusual$ Archimedesestimated the universe !as "222"222"222"222"222 m in diameter" !hich is the

aroximate distance light travels in (+(

years$

In scientific notation" he estimated the diameter of the universe !ould be$ x '2'- m$

In general" you can define scientific notation as follo!s&

Examples:

6rite each of the follo!ing numbers in scientific notation$

'$ '2"222 8 '$ x '2. The exonent of '2 is .$

$ 00"222"222 8 0$0 x '2/ The exonent of '2 is /$

$ +2"222"222 8 +$ x '20

+$ "222"222"222 8 x '21

ote the attern in !riting a number in scientific notation$ The decimaloint !as moved to the left so that the multilier !ill be a number bet!een ' and

Scientific otation

A number is in scientific notation if it is exressed as aroduct of t!o factors" one factor being less than '2 andgreater than or e*ual to 'and the other a o!er of '2exressed in exonential form$

++


12/23

'2$ The number of laces !ill be the exonent of '2$ If you move to the left" theexonent is ositive$

Ho!ever" if the decimal oint is to be moved to the right" the exonent !illbe negative$

.$ 2$222- 8 - x '25+

-$ 2$22222222/1 8 /$1 x '251

Note: To convert bac% to standard or decimal form" the rocess is simlyreversed$

Examples:

6rite each of the follo!ing scientific notation in standard form$

'$ $' x '2.

$ ' 2 2 2 2

Therefore" $' x '2. 8 '2"222 in standard form

$ 0$. x '21

0$ . 2 2 2 2 2 2 2 2

Therefore" 0$. x '21 8 0".22"222"222 in standard form

$ -$ x '25

2$ 2 2 -

Therefore" -$ x '25 8 2$22- in standard form

+$ $0 x '25-

2$ 2 2 2 2 2 0

Therefore" $0 x '25- 8 2$222220 in standard form

The exonent is ositive . so you move thedecimal oint . laces to the right$

The exonent is ositive 1 so you move thedecimal oint 1 laces to the right$ Add #eros if

needed/

The exonent is negative so you move thedecimal oint to the left$ Add #eros if necessary/

The exonent is negative - so you move thedecimal oint to the left$ Add #eros if necessary/

+(


13/23

Scientific notation is useful in exressing large and small numberssecially !hen you are solving statement roblems$

Examples:

Solve the follo!ing roblems$

'$ >ight travels at a seed of $2. '20 meters er second (m:s)$ Thereare aroximately $'. x '2/ in a year$ Ho! far does light travel in ayear<

Solution&

Multily the distance traveled in ' sec by the number of seconds

in a year$

This gives ($2. x '20)($'. x '2/) 8 ($2. x $'.)('20x'2/)

8 1$-2/. x'2'.

Fou multily the coefficients and add the exonents$

$ The distance from earth to the star Sica ( in Girgo) is $ x '2'0 m$Ho! many light5years is Sica from earth<

Solution&

+)+'

+)

+'

+,(/(+,

+,(/( −

= x

x

8 $ x '2

8 2 light years

Try this out

A$ 6hich numbers are in scientific notation<

'$ 2$1 x '2

$ '$- x '25.

$ '2$- x '2

+$ 0$- x '2+

.$ 2$+ x '25'

7ivide the distance (in meters) bythe number of meters in ' light5year$

+.


14/23

Each of the standard numerals is given in scientific notation$ The scientificnotation may not be correct$ Identify the numbers that are correct$

-$ +/"222 8 +$/ x '2+

/$ '"+22 8 $'+ x '2+

0$ ."-2+"222 8 $.- x '2/

1$ +"-2"222 8 +$- x '2/

'2$'/1"112"222 8 '$0 x '20

3$ 6rite in scientific notation$

'$ '2"222

$ 2$222/0$ 2$'-.

+$ '.

.$ -/0"+."0+0

-$ "/0+"-22

/$ 2$2222222.+

0$ 2$222222//

1$ .$ x '2.

'2$ 1-+2222 x '25

C$ 6rite the follo!ing in standard form$

'$ The roc%et is $0 x '2. %m above the earth$

$ The satellite travels + x '2 %ilometers er minute$

$ Sound travels 1 x '2+ meters er minute in !ater$

+$ 6ater has a !eight of ' x '2'' %ilograms er cubic %ilometers$

.$ There are about $ x '20 molecules in an atom$

7$ Solve the follo!ing roblems$

-$ The diameter of a hydrogen atom is 2$2222222' cm$ Exress thediameter in scientific notation$

+-


15/23

/$ The mass of the earth is about ./1. 222 222 222 222 222 222 %g$Exress the mass in scientific notation$

0$ The farthest obect that can be seen !ith the unaided eye is the Andromeda galaxy$ This galaxy is $ x '2 %m from the earth$ 6hat is

this distance in light5years<

1$ A light year the distance travels in one year is e*ual to 1$++ x '2' %m$If the =olaris is about -+ 222 222 222 %m from the earth" ho! long !illit ta%e the light from this star to reach the earth<

'2$The seed of radio !aves is 1/ -22 %m er second$ Ho! much timeis needed for the radio imulse to travel from a station to a radio if thedistance bet!een them is '02 %m<

Let’s summarize

'$ 9or any real numbers a" and integers m and n" the follo!ing la!s of exonentalies to exressions having the same base&

a$ Multilication&am , an 8 am ? n

b$ =o!ers of =o!er (am)n 8 amn$

c$ 7ivision&am an 8 am 5 n

$ Bero Exonents&

9or any real number a !here a ≠ 2" a2 8 '

$ egative Exonents&

9or any non#ero real number a and !hole number n" nn

aa

+

=

−

and a5n isthe multilicative inverse: recirocal of an$

+$ Exonential E*uations are e*uations such that the un%no!n is an exonent$

.$ Scientific otation

+&


16/23

A number is in scientific notation if it is exressed as a roduct of t!ofactors" one factor being less than '2 and greater than or e*ual to 'and the other a o!er of '2 exressed in exonential form$

What hae you learned

A$ Simlify the follo!ing exressions$

'$ 0.2 -$ +!5

$ 40t2 /$ (+!)5

$ (m+n)2 0$*

&

−

−

d

c

+$

,

,

+)b

a+

1$-

+− x

.$ (5)()2 '2$ (

-.( 12−

−

b

ba

3$ Simlify the follo!ing&

'$ (.)+ -$ (+a.)(.a)

$ . , . /$ (.!5)(+!)5

$ (4+t) 0$ +.x/y+ .xy

+$ (m+n) 1$*

&

&

(,−

−

c

c

.$ (5.)() '2$ y x

y x(

-.( 12−

−

C$ Solve for x$

'$ 0x ? 8 02 -$ -x 4 4 . 8 '1'

$ -x 4 + 8 -'- /$ '-x ? ' 8 +x 5

$ .x 8 + 0$ 0x 8 '-x 5 '

+)


17/23

+$ /x 8 0' 1$.

+(

(*+ x

80'

.$ .x 8 -. '2$ +(&(&.

(

+

=+ x

7$ 6rite in scientific notation$

'$ '/2"222"222

$ 0"22"222"222

$ 2$222222.

+$ 2$2222222222-.

E$ 6rite the follo!ing in standard form

.$ +$22/ x '20

-$ $/1' x '25+

/$ +$/.- x '25.

9$ Solve the follo!ing roblems$

0$ The mass of the sun is aroximately '$10 x '22 %g$ If this !ere!ritten in standard or decimal form" ho! many #eros !ould follo! thedigit 0<

1$ Megres" the nearest of the big 7ier stars is -$- x '2'/ m from earth$ Aroximately ho! long does it ta%e light traveling at '2'- m:year" totravel from Megres to earth<

'2$ The number of liters of !ater on earth is '." .22 follo!ed by '1 #eros$6rite this number in scientific notation$ Then use the number of liters of !ater on earth to find out ho! much !ater is available for each ersonon earth$ The oulation is .$ billion$

+*


18/23

!"#WE$ %E&

Ho! much do you %no!

A$ '$ 0' -$ x0

$ -. /$ '2a/b

$ /-0 0$ -abc

+$ 0 1$ +x/y.

.$ 1 '2$ +x+y+

3$ '$ ' -$ 5'

$ ' /$ 1

$ . 0$ '

+$ ' 1$ '

.$ / '2$

C$ '$ .

+

-$

-

-

x

y

$ +,,

+

/$--

+

y x

$ &+(

+

0$ -+

+$ ' 1$.

.

x

y

.$.

.

x

y

'2$)

)

a

b

7$ '$ x 8

$ x 8

+'


19/23

$ x 8

+$ x 8

.$ x 8 -

-$ '$0+-0 x '2/

/$ '0"+.2"222"222"222"222"222

0$ x '2. %m:sec

1$ -$2 x '2.

'2$ -$+ x '2'

Try this out

>esson ' A$ '$ '2+ -$ -+

$ '. /$ +'112+

$ 1' 0$ '1-

+$ '22 222 1$ .'

.$ ///- '2$ ' .22

3$ '$ +21- -$ b2

$ +1 /$ '-b-

$ .- 0$ '.x'.y1

+$ -+ 1$ '-c'2d-e

.$ . '2$ 1a-b+c

C$ '$ '22 -$ r

$ 0' /$ /a'.

$ -. 0$ 1xy+

+$ / 1$ x2y

+3


20/23

.$ .- '2$ -a+bc2

>esson

A$ '$ ' -$ y

$ ' /$ '

$ 0 0$ '

+$ 4/ 1$ '

.$ '2$ 5.

3$ '$+,

+

a -$&

+,

x

$ +)

+

/$-

+)

+

y

$ 5+)

+

0$(

&

t

−

+$ (&

-

1$ x+

.$-

.

w '2$ .

( (a

>esson

A$ '$ x 8 + -$ x 8

$ .

+= x

/$ x 8

$ x 8 5 0$ x 8

+$ x 8 + 1$ x 8

.$ x 8 '2$ x 8 5.

(


21/23

3$ '$ x 8 -$ x 8

$ x 8 5' /$ x 8 ''

$ x 8 / 0$ x 8 '

+$ x 8 '2 1$ x 8 5

.$ x 8 + '2 x 8 1

>esson +

A$ '$ o -$ Correct

$ Fes /$ Correct

$ o 0$ 6rong

+$ Fes 1$ 6rong

.$ o '2$ 6rong

3$ '$ '$ x '2 .

$ /$0 x '2 5+

$ '$-. x '2

5'

+$ '$. x '2

.$ -$/0+.0+0 x '20

-$ $/0+- x '2-

/$ .$+ x '250

0$ /$/ x '25/

1$ $. x '2-

'2$ $1-+ x '2+

C$ '$ 02"222 %m

$ +2 %m$

(+


22/23

$ 12"222 m

+$ '22"222"222"222 %g

.$ 2"222"222 molecules

7$ -$ ' x '250

/$ .$/1. x '2'

0$ "22"222

1$ $+/ days

'2$ +$2'' x '25

6hat have you learned

A$ '$ ' -$(

-

w

$ 40 /$(+)

+

w

$ ' 0$&

*

c

d

+$ '/ 1$ x+

.$ 40 '2$)

)

a

b

3$ '$ -. -$ 2a0

$ '. /$-+)

&

w

$ 5-+t 0$ 1x+y

+$ m'n- 1$ +c

.$ ''. '2$-

.

x

y

((


23/23

C$ '$ x 8 -

$ x 8 '2

$ x 8 '

+$ x 8

.$ x 8 '

-$ x 8 -

/$ .

&−= x

0$ x 8 5

1$ (

+= x

'2$ .

+= x

7$ '$ '$/ x '20

$ 0$ x '21

$ $. x '25/

+$ -$. x '25''

E$ .$ +22"/2"222

-$ 2$222/1'

/$ 2$2222+/.-

9$ 0$ 0

1$ -- years

'2$ '$.. x '2 and $1 x '2'

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Module 1 Integral Exponents (1)

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