## Note on Law of Indices

• Note
• Things to remember
• Exercise
• Quiz

Indices is a number with the power. For example: am; a is called the base and m is the power. These laws only apply to expression with the same base.

Index help to write a product of numbers very compactly. Index help to show how many times to use the number in a multiplication. It is shown in the top right of the number in small number.

In this example: 4³ = 4x4x4 = 64

### Rule1: a° = 1

Any number, except 0, whose index is 0 is always equal to 1.

An example:

2° = 1

### Rule 2: a-m = $$\frac{1}{a^m}$$

An example:

2-3 = $$\frac{1}{2^3}$$ ( using a-m = $$\frac{1}{a^m}$$)

### Rule 3: amx an = am+n

In case of multiplication of same base, copy the base and add the indices.

An example:

3x 34 = 32+4 (using am x a= a m+n)

= 36

= 3 x 3 x 3 x 3 x 3 x 3

= 729

### Rule 4: am ÷ an = am-n

In case of division of same base, copy the base and subtract the indices.

An example:

w10 ÷ w6= w10-6 = w4

### Rule 5: ( am)n = amn

To raise an expression to the nth index, Copy the base and multiply the indices.

An example:

( x2)4 = x2x4 = x8

### Rule 6: a$$\frac{m}{n}$$ = ($$\sqrt[n]{a}$$)m

An example:

125$$\frac{2}{3}$$ = ($$\sqrt[3]{125}$$)2 = (5)2 = 25

• An indices is a number with the power.
• The laws of indices state a number of rules, which can be used  to simplify expressions involving indices.
• Any number, except 0, whose index is 0 is always equal to 1. (i.e. a° = 1)
.

### Very Short Questions

Solution,

(x3y)×(xy)×(x2y)

= x3× x× x2×y×y×y

=x3+1+2×b1+1+1

= x6y3

Solution:

$$\frac{-36a^8}{9a^5}$$

=$$\frac{-4×9a^{8-5}}{9}$$

= -4a3

Solution:

(a2b)×(ab)

=a2×a×b×b

=a2+1×b1+1

=a3b2

Solution:

(-7p3)4

=(-7)4.(p3)4

=74.p3×4

=74p12

Solution:

(xy2)3×xy

=x3(y2)3×xy

=x3y2×3×xy

=x3.x.y6.y

=x3+1.y6+1

=x4y7

Solution:

(4x4)×(3x3)4=43(x4)3×34(x3)4

=43.x4×3.34.x3×4

=64×81.x12.x12

=43×34x12+12

=43×34 x24

Solution:

$$\frac{(3p^2q)^2}{p^2q^2}$$

$$\frac{3^2(p^2)^2.q^2}{9p^2q^2}$$

=$$\frac{9p^4.q^2}{9p^2q^2}$$

=p4-2q2-2

=p2.q0

=p2

0%

5p
7p
10p
6p

83
87
85
88

6x5
6x8
6x7
6x4

x2
x2y
7xy2
x2y2

7
2
3
5

am+n+2
am+n+4
am+n+1
am+n+3

a3c.b3c
a2c.a2c
a5c.a5c
ac.ac

11
8
9
7

-5a3
4a3
-4a3
5a3

-5xy2
-3xy2
3xy2
-2xy2
• ### Find the value, by using the law of indices:(4x4)3 ( imes)(3x3)4

43( imes)34 x25
45( imes) 34 x24
43( imes) 34 x24
42( imes)33 x24

3x2
7x2
5x2
4x2

a6b3
a7b3
a3b3
a5b3

56
55
58
57

x13
x12
x15
x10

437, 500
435, 550
436, 545
430, 540

x=8
x=7
x=5
x=4
• ## You scored /17

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