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Mathematics (Algebra): Solved Questions on Exponents
Law of Exponents
(1) am × an= am+n
(2) am ÷ an = am--n
(3) (am)n= amn
(4) (a1/n) n = a1/n ×n = a
𝑎

𝑛

𝑎

(5) ( )m/n = ( √ )m
𝑏

𝑏

(6) a0 =1
(7)

1
𝑎

= a—1

𝑎

𝑏

𝑏

𝑎

(8) ( )—1 = ( )

Exercises
(1) 323/5
5

( √32)3
23 = 8

(2)

2x = 0
...
5 = 2515/10 = 253/2
( 2√25) 3 =
53 = 5x5x5=125

5
40

=

1
8

= 2--3

(4) 4/2x = 64x
22/2x = 28x
22-x =28x
2-x=8x
2= 8x+x
2= 9x
9x =2
𝟐

X=

𝟗

(5) 10X =

1
0
...
125)2/3 =(−
125 2
)
1000

3

(√−
5

1

10

2

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

1

2

2

=-- X-=

𝟏
𝟒

(7) (0
...

32x= y2

y2 –9=0
(y+3) (y—3) =0
Y+3=0 or y—3=0
Y=--3 or y=3
Y=--3, 3x=--3(no solution)
When y=3, then 3x=31
X=1

(10) 42x—21x4x+80=0
Let 4x be y, then y2 –21y+80=0
By factorization, (y—16)(y—5)=0
Y=16 0r y=5
When y=16, then 4x=16
4x=42
X=4
When y=5, 4x=5
Take the log of both sides, xlog4= log5
X=

𝑙𝑜𝑔5
𝑙𝑜𝑔4

Log5—log4
= 0
...
6021
= 0
...
250
So, x= 2 and x= 1
...

32

5

(√−

243 3
)
32

3

3

3

2

2

2

(− )3 = -- 𝑥 --

(11) 4—4x =

𝑥−

3
2

= -- -

𝟐𝟕
𝟖

= -- 3-

𝟑
𝟖

1

256

4—4x =256—1
--4x=--4
X=

−4
−4

X= 1

1

(12) 16(4)2x = ( )x
2

24 x 2(2)2x = 2—x
24+4x =2—x
4+4x= --x
4x+x=--4
5x=--4
X=-- -

𝟒
𝟓

1

(13) ( )(2-y) = 1
4

4—1(2--y) = 1
4—2+y = 1
4y—2= 40

4—4x =4—4

y—2=0
y=2

(14) 21/2 x 81/2/4
21/2 x (23)1/2/22
21/2+3/2—2/1
21+3—2/2
24-2/2
22/2
21
=2

(15) 3(1—n)/9—2n=1/9
3(1—n)/32(--2n) = 1/32
3(1--n)/3—4n =1/32
31—n-- --- 4n =3—2
31—n+4n =3—2
31+3n = 3—2
1+3n= --2
3n = --2—1
3n= --3
n = --1



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