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PATH TO SUCCESS
ON
INDICES AND STANDARD FORM
(WITH QUESTIONS AND ANSWERS)
BY JOHN EKPENYONG
Objectives
At the end of the study note students should be able to :
Define indices
Solve problems related to indices
...
Solve problems on standard form
...
INDICES
Indices is the power or exponent which is raised to a number or a variable
...
LAWS OF INDICES
The first index law : am x an = am+n
: Example
Second index law : am/an = am-n
: Example
Third index law : a0 =1
Fourth index law
(am)n
a6 x a3 =a9
a5/a2 =a3
or
or
a(6+3)= a9
a(5-3) =a2
(in index law any number raise to power zero is 1)
:
Example (a5)3= a15
Fifthth index law : (axb)m =amxbm : Example (5x3)5=55x35
(a/b)m
=am/bm Example (5/6)3 = (5/6) x (5/6) X (5/6)
=(5x5x5)/ (6x6x6)
= 53/63
Negative powers x-a
= 1/xa
OTHERS INVOLVING ROOTS
√a = a1/2 Example √4 = 41/2 =√4=2
3√x =x1/3 Example 3√27 = 271/3 =3√27 =3
Example 1
Multiply (a) x5 by x3 (b) a3 by a2
(c) y by y4
Solution
Method 1
x5
(a)
x x3 = ( x × x × x × x × x)
= x×x×x×x×x×x×x×x
=
x8
Method 2
X5 × x3 = x
5+3 =
x8
× (x×x×x)
(b)
a3 x a2
Method 1
a3 x a2
= ( a x a x a) x ( a x a)
=axaxaxaxa
=
a5
Method 2
a3 x a2 = a
3+2
= a5
(c)
y x y4
Method 1
y x y4 = ( y) x ( y x y x y x y )
=yxyxyxyxy
= y5
Method 2
y x y4 = y1+4 = y5
Example 2
Simplify (a) 10-3
(b) 12x7÷4a3
(c) (1/4)-2
Solution
(a)
10-3 = 1/103= 1/1000
(b) 12a7÷3a3 =12 x a x a x a x a x a x a x a
(3xaxa)
4 x (3xaxa)a x a x a x a x a
(3xaxa)
=4 x a x a x a x a x a = 4a 5
(c) (1/4)-2 =1/(1/4)2= 1/(1/16) = 16
STANDARD FORM
STANDARD FORM
The standard form of number is a way of writing the number in a form that
follows certain rules
...
For example : 1000000 = 10 x 10 x 10
x 10 x 10 x 10
Example 1 :
Express the following in standard form
...
5 x 100 = 6
...
8 x 100000000 = 4
...
(a)4 x 104
(b) 4
...
8 x 107
Solution
(a) 4 x 104 = 4 x 10000 =40000
(b) 4
...
3 x 10000 = 43000
(b) 7
...
8 x 10 000 000 = 78 000 000
Example 3
Express the following fractions in standard form
...
00007
(b) 0
...
000 000 022
Solution
(a) 0
...
075 = 7
...
5 x 10-2
(c) 0
...
2/100 000 000 = 2
...
(a) 9 x 10-4
(b) 9
...
0009
(b) 9
...
4 / 10000 = 0
...
1
...
Solve (1/4)-2
[a] 16
[b] 4
[c] 1/2
[d] 1/8
3
...
000 006 3
[a] 6
...
3 x 106
[c] 6
...
3 x10-7
4
...
Solve p6 ÷ p3
[a] 1
[b] 14
[c] 6
[d] 15
6
...
[a] 9 x 103
b) 9 x 104
c) 9 x 10-4
[d] 9x 10-5
7
...
3 x 10-7 as decimal fractions
...
000 0053
[b] 0
...
000 053
[d] 0
...
Solve m0 x n0
[a] 2
[b] m + n
[c] m0
[d] 1
8
...
Solve 2a-1 X (3a)2
...
Simplify (1/3)-2
...
Express 56000 in standard form
...
6 x 102
[b] 5
...
6 x 104
[d] 5
...
Express 53 in standard form
...
[a] 27
[b] 22
[c]
28
[d] 29
Simplify (24)2 in indices form
...
What is the value of 10 (-2) in standard form ?
[a] 0
...
00001
[c]
0
...
0001
Answers
1
...
A
A
3
...
A
5
...
B
7
...
D
9
...
D
11
...
C
13
...
C
15