_,, ____

XPONENTS

to study exponents?
d O youneed
Y arth . 5 976 000 ooo,000,000,000,000,ooo kg.
0 fE is ' , ,
. 000,000,ooo,ooo,ooo,ooo,ooo kg.
ofuranus 15 86' 800'
ch haS a greater mass, Earth or Uranus?
make these large numbers easy to read, understand and compare.
entsare used to

PONENTS
know that 10 + 10 + 10 + 10 + 10 = 50 whereas 10 x 10 x 10 x ~0 x 10 = 100000.
10 + 10 + 10 + 10 can also be written5
as 5 x 10 and r:ad as 5 times 10.
10 >< 10 >< 10 x 10 is written as 10 and read as 10 raised to the power 5.
es,'10' is called the base and '5' the exponent (or index or power).
is called the exponential form or power notation of 100000.
ly, 11 + 11 + 11 = 3 x 11 read as three times 11 and 11 x 11 x 11 = (11)3 = 1331 and is read as 11 raised
power 3. In (11)3, 11 is called the base and 3, the exponent.
al, if xis any integer, then x multiplied by itself repeatedly n times given as x x x x x x x ... n times
Is denoted as r'. Here xis called the base and n, the exponent or the power or the index.
uct of 'n' factors each of which is x.
pie, (7)6 = 7 x 7 x 7 x 7 x 7 x 7 and (2)4 = 2 x 2 x 2 x 2.
exponents are a "short cut" to show that a number is to be multiplied by itself a given number
1he operations x2 (read as x squared) and x3 (read as x cubed) are examples of exponents. The
forms get their special name from the square and cube of Geometry. The area of the square
% x z or iJ.. The volume of the cube with side x is x x x x x or x3 • Thus, we call these 'x square'
~~~ . --

two simple rules of 1.

to the power of 'one' equals itself.
power shows how many times the base occurs as a factor.
15 and (10)1 = 10.
1111N111~ is one. This is so because one multiplied by one as many times is always equal

1 >< 1 >< 1 x 1 = 1; (1)2 = 1 x 1 = 1 and (1)205 = 1.

(-1)2 = (-1)(-1) =1 (-1)3 = (-1)(-1)(-1) =-1

(-1)5 • (-1)(-1)(-1)(-1)(-1) = -1
(-l)ewn number • 1

(-1,., • -1 3. (-1)270 • 1 4. (-1)562 = l

 Exercise 5.1 _Cl
ollowing table.
Expanded Form Value
al Form Base Exponent
4 3 4x4x4 64

~ 3 ,(i -,( 3 :x :) it \
(5)5 ~ ~ A. ( x( X ~X\ 31".l.)
7
()
:r =,-
(2)2 '2- 2....X.-2-. ~
•• ress the following in the exponential form.
8XP 4 4 x 4 x 4- -9 x -9 x -9 x -9
b c. axaxaxaxa
' )( 4 )(x 3 )(x 2 x 2 x 2 x 2
. . '!II
~e --b x -b x -c x -c x -c f. f X f X f X U X U X V

d, .i >< 3
find the value of 4 e. (-17)2
{3)' 'Aof c. (12)3 d. (-5)
: entity the greater number in each of the following pairs. 3
100 5
~ )5 or (5)2 ~ 3 or (-3)7 ~ )10 or (10)2 ,.d,{100)2 or (2) e. (-10) o~0)

Simplify:
b. 33 x (2)2 C. (5)2 X (4)2 -...,tY.t) X (12)2 X (13)5
L 9 >< 105
Find the results.
L (-6)' b. (-4)2 x (-3)3 _¢)2 x (-8) d. (-10)4 X (-5)2
• Express the following as a product of prime factors.
L 49 b. 625 c. -512 d. -216 sA132 __V-6075
Simplify the following.
1
a. (9-32) ic..(2)3 /(7'2 + 42) X (3)2 C. (32 - 22) + -
25
Bq. . the following in index form.
b. 343 c. 1331 d.243 e.3125

RUW OF EXPONENTS
g Powers v;ith the Same Base
c.kulaka th(• following.
b. (- 5f x (- 5) c. a5 x J➔
~•-• Px3x3x~xpx3x~=3x3x3x3x3x3 x3
· • (3)"
~,-(-5 • -5) • (-5) • -5 • (-5) x (-5)
(-5,S
• M • IIC • JC ti) JIC (• ,c 11 ,c ti ,c ti) • ti ,c ti IC II X a JIii: 8 X a X ti M ti X ti

~ts @
r

power notations in final

··; Now, carefully study the Incorrect
ps in pa rts a, b an d c of the above example. You
ste
the RHS is the sum of tlle
will see that the index on
indices on the LHS. The
. · . . d m and n are whole
an
Thus, if x 1s any mteger
numbers, then xm X xn =
xm+n
, r rn+ n+p+q+r
Similarly, x'!' x xn x xP x x9 x x =x
3 2 5 -- -- -- _,____~ illlcl~
)2 = (l1 ) + = (l l)
For exam ple, (11) x (11
3
(11)21
9+ 5+ 4+ 2+ 1 --
(11) 2x 11 = (11 )
and (l1)9 x (11)5 x (11)4x

Ru le II: D iv id in g Po w
er s w ith th e Sa m e Base

AM rL E 7: So lve the
Ex
following .
(- 4) 6 x4
35
b. (- 4)3 c. x2
a. 32
JXJX3X3X3 b. (- 4) : = (- l)x (- ,( )> <( -
35 (-,4') ...
S o t.UTIO N: a. 32 = ,.3' X t (- 4)

35 (-4)6 = (-4)3
2 =(3)3 (-4)3
3.
x4
4 . 2 = x2
x ;t x; tx xx x y non-zero mteger x
c. 2 = - - - - , for an X
X ,:( X; (

d c of th e above exa:At~e
.
tations ob tai ne d in pa rts a, b an
Now study the power no s on the ~
in de x on the RH S is the difference of indice
You will find that the
ro integer x
In general, for any non-ze Remember
xm +x n =x m- n
e numbers and m > n. If x is any non-
where m an d n are whol
11 =(9)11-7 =(9)4 w ho le nu m be rs,
For example, (9) + (9)7

_ _ _ e S
~ .
~2'.___-:.:!!
11_ _ _ -- =~
E ~
xe ~
ra·s=
Simplify the following.
r a. (2)9 X (2)ll = _

C. (]5)2 X (15) 3 X (J5) 4 = _ d. (-7)5 x

e. 1011 +
-
104 = f. (-7)2:5 +
g. (5)16 + 5 =
- h. (-3)41 .
i. (z)90 + (z) 80 =
- j. (-p)1~ .

~ ICSE Mathematics 7
 5)J _~ X:_ X-;_ -;:; 0 X7 l' 1
1. (- - 7 7 7 )( 2

7- )2 -3 -3 _ -
-3 x -3-;:; - x- - 8
(-3J umber, then
( 3 -;:; _:..--:::- 8 8 hole n
2. s' 8 >< 8
. ers Y:;t O
and ttt is a w
. d y are lllteg
fhUS, 1f X all ( J,n
x)m - x
(~-
(yt y
-
. Exponent tural number, then r"' • t
le VII: Negatrve d ttt is any na -s
Ru . nurnber art
- ero rat1ona1 .
If x is any non z o -m
1 X ::: Xo-111 ::: X
Observe that -; == ?
x . examples:
Consider the following
1 1
I . 2-J = -23 = -8

3-2 42 = !§.
.l ?=J2 9

r. ..

,.,
a.
\\t l'L~

Gf
11: Find the value of
b. (- : r
2
_3 1 1 _f_= 3x3x3 =E
Soll TI0'1: a. ( 3) : (2 )3 = t - z3 2 2 2 8 X X

3 33
5)-2
(-iJ
1 ( 7)2 (-7)2 -7X(-7) - 49
b. - 7 ( = = - 5 = (5)2 = 5 X 5 . - 25

3
)-3
c. ( -11
4
= (- ~J -
1

1
-(- 11] _ (-11)3 _ -11 X (-11) X (-11) _
4 - 43 4x4x4

I Exercise 5.3
l Simplify and write the answer in exponential form.
a. (82)7 = _ b. (33)100 = _
d. (2_.y19 = - e. (-53)90 = _

70 JCSE Mathematics 7
 li£y the following and express the answer with a single exponent.
2. Siil'P)5 x ( )5 = _ _ .b. (2)7 x (a)7 = _ _ c. p9 x q9 = _ _
4
a. (3 5- e. (-a)6 + (-b)6 =- - £. 4,r- + 9q2 =
d. 3s-1-S - - --

(-5y)
Siil'Plify:

. (tJ b. (ff c. 7x
3

.. 4-3
1
b • -5-2 . c. (-6t3 d. (ff e. (-7)-2

5unplify: c. (3x2)2(xyt2
b. (-4y)3
a. (-3%)2

WOf EXPONENTS FOR RATIONAL NUMBERS

laws or rules of exponents on integers can be extended to rational numbers. Thus, for any rational
bers!!.and !.., and positive integers m and n,
q

(:)' ={
s

b. (: rJr
X (: = (: t (:J +(:J =(:r
{(!r}" =(:f e. (: J= 1 f. (: Jx(ff xff = (:

liI
(ff
(pxs)m
s

qxr

12: Simplify the following and write the answer in the exponential form.
5
b. [(- 3) + (- 3) ] x(-3)
3 7
c.[<9-2r• x(: )}(¾)'
7
28 xa6
4 10 6
e. (7 X 7 ) X (9) £. 43 X a4

(8' ,c 8') + (S2)3 =(8)3+9 + (8)2><3 =(8)12 + (8)6 =(8)12-6 =(8)6

X (-3)1 • (-3)5-3 x (-3)7 = (- 3)2 x (-3)7 =(-3)2+7 =(-3)9

---•QJ- [(9)-2•➔ x(: )}G)" 1

= [(9)" x(:1 )}(:)" = (9x ;1 Jx(! J
•~· x-}Jx(!)" = GJ" x(! rs(: x :r 8
=(1) =1.

Exponents @
~
-
.. c-Sirnplify (25)"1 + (-5t 1
/1,J•
25 4 + (-5t7 = (52)4 + (-5 )-7 = 52><4 --,-. (- 5)-7
LtJTIO,r. ( ) = 58 + (-5t7 = (-5)8 + (-5)-7 (·.- 5 8 = (-5)8, as 8 is an even number)
= (-5)8-(-7) = (-5)8+7 = (-5)15
105
4 6
'"' 16: sunplify 2 x 5
5
10s _ (2 X 5)6 = 2s55 25-4 2
tUTION: i' X s6 - 2
---...,.. 4
X 5 2
X
4
X 5
6 56-5 =s
. d p so that (-5)s x (- 5)7 = (- 5)3P.
LI 17: Ftn
(-5)8+7 = (-5)3P
LUTION: (- 5)8 X (-5)7 = (-5)3P ⇒
(- 5)15 = (-5)3P
(Since the bases are same, powers can be equated.)
15=3p
15 p=S
p=-
3
,u 18: Solve for x if 32x-1 = 92 x 27.
32.x-l = (32)2 X 33 . ⇒ 32x-1 = 34 X 33

32.x-1 = 34+3 ⇒ 32x-1 = 37

8
2x-1 =7 ⇒ 2.x = 8 ⇒ x=-=4
2
· x• 4
LB 19: Which is larger?
3 x 108 or 28 x 107
Make the powers of 10 same in both.
10) X 107 = 30 X 107
3 )( 108 • (3 X ⇒ 30 X 107 > 28 X 107
.". 3 X 108 > 28 X 107

Exercise 5.4 .-~I I

the following and express the answer in exponential form.
7' x 1J >< ~ b. (8)15 + (8)12 c. [(11)4]5
(5)' X (5)' + (5)2 e. p X q4 X r4
4 f. (225 + 220) X 210

IT ·GJ h. (10"')10 + (10)90 i. (-6)• + (-6)

k. (- 7)0 + (8)0 + (S)o l.

that ,i, x S2 • fJS? lf not, why?
tie numerator and denominator of each of the following as a produc~ of powers of
. , . . , .and then simplify.
8 ><92 x63
. · .._, ·· •
5
.. 9
66 x103 xS 35 x10 x25 25.x25xx
122 )( 35 c. 154 X 83 d. 57 X 65 e. 103 X x5
~~~""'
Exponents @
 '- TN i', bc o- -W 1 r: c :r -. ~ «
s. f ... ., JIO t¥ 1M de ti' '-' ' .._ .. .... . . •"·- "
.. ..
.. . ~ Q.

... ·-· ... .

• ,: i • •
.,
. ~
L Fa!M - - - ? > :\
:! or - ? • 4 .. -4 > - )

-~ .. :
• • ! "' -~
5 "
7. fu c. li ll a -: •~ ' • .. ..
ll • •

-
•l m o n ~ h ln ~
l :'.'11
d li li it o l~ lill\
th r nw nb ff ~ ~
..
4lj'l .- -
ill, ~
a. Tna,r, - 5 .!t ie so nn gb la I
nd :.! on ld t.w vl
-7 7
11
as 0, u -. 'i=:r l t
-.,... .....
~
4' ',1.

,~
E. ur cis e 5.1
~-
.
u ... .!.. I

··~
L up rm ti al 6l ~ "~w .. .<' l
...
a.
(m a

(4)3 4
¼ ' mt
3
E; Va lai l u. ... =-u ~~
., ;I
..
r.:

b. • · -4 • 4 61 \-4. ~ - 9 . ,: .. ,. ~ - C
(3)4 3 4 ,,' 1 'l "
3• 3• 3 •3 11
C. (5)5 5 5 llc ,"li..
5• 5• 5• 5 • 5 3U 5
cl. 7 7 1 7 \. a.. l, 5
(2)2 2 7
2
2. a. 46 b. (-9)4 C. ,,,
2• 2
• 2. a. S
• -:_,i, C \,! '
d.J lx' .24 e. (-b )2 x (-c )3 f. tlulo
l. L S\
.. ~ ' ~

3. a. 72 9
4. a. 2s
b.1 00 00 00 0
b. (-7)3
c.1728
C. 210
d. QS t.2 89
4. l! \
.- l ~

5. a.9 00 00 0 cl. 2 100 t. U) l

b. 1~ c.4 00
6. a. -2 16 d. O <. !'- •'· ~
7. a. '12
d. (-6 )3
b. -4 32
b. 54
e.' .24 •7 •1 1
c.- 39 2
C. (-2 )'
f. (- 5)1 • (- 3)5
d.250000

6.& . 9
cl. ~
) " UH
~- :..,
8. a.1 28
9. a. 2 5
b. 58 5 c.1 25
d. 48
b. I
c. mI cl. ~
7. iL l"8
Exerc ise 5.2
b. 73 C. U l
d. 35 e. ss
b. 719 c. J? d. lo ~ -....
~
t
a. 2 20 b. (-3 )7 d. 'S C, ~
f. (-7 )6
C.(15)9 d. (-7 )16 i '~
g. 51s e. 107
h. (-3)17 i. z10 10 . iL 16 b. )
Ex erc ise 5.3 j. (-p )25 (. 5
Ob 1e ch ~ c fy rc Q
1. a. g2 x 7 = gl4 ue ,u
b. 33x 100 = 3n A. 1. iL (-4 )u b .
d. 2-4 •-1 9 = 16 C. S25 . 7 - (-6 f c. (-2 )\~

.'
2 e. (-1)9() (5)3•9Cl ~ 5175 d. ~
s270 'l. 0
2. a. (3 x 4)5 = 125
b. (2 x a)7 -(2 a)7
f. (-11)4 •25 • (-11)1
00
4. 2 s. 5 111 6. l
).
(. (p X q)9 • (pq)9 10. ..!._
7. - 64 s. ..!... :
~ ~"
d. (¾J f. (2p )2 =(2p) B. 1. Fa lse
64
11 .
s 12 . l

(3q)2 2. Fa lse
3. a . ~ ~3 q 5. Fa lse 3. Tr ur
b. 25 6. Tr ue
256 -1 25 y 3 Exerc ise 6.1 7. Tr ue
4 C . - --
1 1 343x3
4. a. 43 = 64 1. a. No

e. _l _= ..! ._ 5. a. 9x2
c. (~ )3 = ;116 d. (½ = :7
2
J 2. a. Tr ue
3. a. e
b. Yes
b. Fo lse c. Tr
b. II!
c. No

c. i!
ue
d. No
d . False
b. -6 4y 3 d . i!
(-7 )2 49 c9x
.- f. e g. e h. II!
e. •
y2 4. a. Ro ste r fo rm
Exerc ise 5.4 : (19, 21, 23, 25, 27,
Se t Bu ild er fo rm : (x 29 1
: x • 211 -1 , 10 S ti
1. a. 714 b. 83 b. (e, x, p, r, s} S I Sl
c. 1120 d. 56 c. (x : x = is ev en
e. (pqr)4

i. (-6 )8 or (6)8 j. a200

£. 21s
g. (~T3 h. 10110
d. (a, e, i}
e. Ro ste r fo rm : (3,
nu mb er, 10 s x s 18}

6, 9, 12}
k. 3 l.0 Se t bu ild er for m: (x
2. No , J,ince 23 x 32 : x = 3n , 1 s n s 4, 11
= 72 an d 65 = 7776 £. (5, 7, 11 , 13 , 17} e NI
3. a. 12 b. 4 g. (x : x = 2n or Sn
c. 9 d. 1 5x4 ; x < 20; x, n e N}
e. -
8 h . Ro ste r M eth od
4. a. x= 5 b. x= 5 c. x= 3 : (0, 1, 2, 3, 4, 5, 6\
d. x = l e. x= 4 De sc rip tio n M eth od
f.x = 1 g. x= 3 h. x= 38 i.x =- 4 : (all wh ole nu mbers
i. Ro ste r M eth od u p to 61
5. a. 26 X 35 b. 29 X 32 C. 22 X 54 : (1, 2, 3, 4\
De sc rip tio n M eth od
25 : {all na tur al nu mb
b. 1 c. -4 j. Ro ste r M eth od ers less than 51
6. a. 36 : {- 3, - 2, -1 , 0, 1,
De sc rip tio n M eth od 2, 3\
: (all int eg ers be tw
een - 4 and 4\
ICSE Mathematics 7