STD VIII-Lecture 17 M.
Prakash Institute
Dear students,
We studied geometry in last 14 lectures and we have seen you en-
joying the proofs of theorems and a variety of problems. Keep it
up!
Now this is the time we will enjoy studying some basics of algebra.
Lets start from some pre-algebra topics.
In mathematics, we consider the following sets of numbers:
1) N = {1, 2, 3, 4, · · ·}, set of natural numbers.
These are also called counting numbers as they are used to count
objects.
2) W= {0, 1, 2, 3, · · ·}, set of whole numbers
3) Z = {· · · , −3, − 2, − 1, 0, 1, 2, · · ·}, set of integers.
Integers are numbers which can be positive, negative or zero, but
have no decimal places or fractional parts.
n o
p
4) Q = q | p, q ∈ Z, q 6= 0 = Set of rational numbers.
Rational numbers are those numbers which can be written as ratio
of two integers. The word rational comes from the word ’ratio’.
Note: Natural numbers are thus termed as positive integers. This
set is denoted by Z + . Whole numbers are termed as non-negative
integers.
There are numbers which cannot be expressed as ratio of two in-
tegers. These numbers are not rational. Such numbers are called
Irrational numbers. For example, we will later learn that square
root of a prime number is irrational.
5) R =Set of real numbers. Set of Real numbers includes rational
numbers as well as irrational numbers.
Indices:
Definition: Let m ∈ N. a ∈ R. Then am means product of 0 a0
taken m times with itself.
i.e. am = a × a × a × · · · m times
Rules of indices:
Let a, b ∈ R, m, n ∈ N. Then
(I) am , an = am+n
(II) (am )n = amn
(III) (ab)m = am .bm .
Question: What is the meaning of am , if m ∈ Z?
So we have to extend the above definition to set of integers.
Definition: Let a ∈ R, a 6= 0, n ∈ Z. Then
1) If n ∈ N, an = a × a × · · · n times
2) If n = 0, an = 1.
1
3) If n < 0, an = a−n .
Do you think the same set of rules will work ? Yes. Definitely.
Rules when indices are Integers
Let a, b ∈ R, a, b 6= 0. m, n ∈ Z.
Then 1) am .an = am+n
2) am ÷ an = am−n
3) (am )n = amn
4) am .bm = (ab)m
Question: What is the meaning of ap/q , where a ∈ R, p, q ∈ Z+ ?
Definition: a ∈ R. p, q ∈ Z+ . Then we define ap/q = (ap )1/q or
(a1/q )p i.e. ap/q = q th root of ap or = pth power of q th root of a
For example,√ √
9
1) x7/9 = √ x7 = ( 9 x)7
2) 253/2 = ( 25)3 = 5√3 = 125 or
253/2 = (253 )1/2 = 15625 = 125
p
We
p
write
√ a q in√several ways as-
a or a or ( q a)p
q q p
Important to Note:
√
1 The symbol is called radical sign. The number written at the
corner of the radical sign is called index.
√
2 2 a denotes square
√ root of a. Many times square root is written
without index as a.
√
3 √
3
a denotes cube-root of a
1
4 n a thus denotes nth root of a and would mean a n .
Exercise: Read out the following numbers / expressions. Find
their values wherever possible.
3 √5
√
5
1) 128 7 2) √ 2432 3) x70
2 8 3
4) y 9 5) m2 6) 625 4
For rational indices too, we have same set of rules of in-
dices:
a, b ∈ R, a, b > 0, q1 , q2 ∈Q
1) aq1 .aq2 = aq1 +q2
2) aq1 ÷ aq2 = aq1 −q2
3) (aq1 )q2 = aq1 q2
4) aq1 .bq1 = (ab)q1 Problems for the Practice:
Evaluate the value (1 to 8 Problems)
(16) −3/4
1 (81)
−4/5
2 (1024)
(243) −4/5
3 (32)
−2/9
4 (512)
5 165/2 ÷ 161/2
22001 + 21999
6 2000
2√ − 21998 √
3
√
81 + 3 −192 + 3 375
7 √3
24
0.25
5 × (125)0.25
8
(256)0.10 × (256)0.15
9 Find x if 33x−5 = 1/9x
10 2h × 23 = 29 . Find h.
Important to Note:
1 Now we know a◦ = 1 for a 6= 0, a ∈ R
2 Any positive power of 0 is 0.
3 00 cannot be determined.
4 In our course, we will consider the roots of positive real numbers
only.
